1.
Orthogonal coordinates
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In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles. A coordinate surface for a coordinate qk is the curve, hypersurface on which qk is a constant. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. The chief advantage of non-Cartesian coordinates is that they can be chosen to match the symmetry of the problem. The reason to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity: many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates many problems may be solved by separation of variables. Separation of variables is a mathematical technique that converts a d-dimensional problem into one-dimensional problems that can be solved in terms of known functions. Many equations can be reduced to Laplace's equation or the Helmholtz equation. The Helmholtz equation is separable in 11 coordinate systems. Orthogonal coordinates never have off-diagonal terms in their metric tensor. These scaling functions hi are used to calculate differential operators in the new coordinates, e.g. the gradient, the Laplacian, the divergence and the curl. A simple method for generating orthogonal coordinates systems in two dimensions is by a conformal mapping of a standard two-dimensional grid of Cartesian coordinates. A complex number z = x + iy can be formed from the real coordinates x and y, where i represents the square root of -1. However, there are coordinate systems in three dimensions that can not be obtained by projecting or rotating a two-dimensional system, such as the ellipsoidal coordinates. More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering their orthogonal trajectories.
Orthogonal coordinates
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Visualization of 2D orthogonal coordinates. Curves obtained by holding all but one coordinate constant are shown, along with basis vectors. Note that the basis vectors aren't of equal length: they need not be, they only need to be orthogonal.
2.
Skew coordinates
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A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates. These coordinate systems can be useful if the geometry of a problem fits well into a skewed system. For example, solving Laplace's equation in a parallelogram will be easiest when done in appropriately skewed coordinates. For this example, the x axis of a Cartesian coordinate has been bent by ϕ, remaining orthogonal to the y axis. Let e 3 respectively be unit vectors along the x, y, z axes. We'll favor writing quantities to the covariant basis. Since the basis vectors are all constant, subtraction will simply be familiar component-wise adding and subtraction.
Skew coordinates
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A Cartesian coordinate system where the x axis has been bent toward the z axis.
3.
Cartesian coordinate system
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In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. The adjective Cartesian refers to the French Mathematician and Philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, the spherical and cylindrical coordinates for three-dimensional space. The development of the Cartesian coordinate system would play a fundamental role in the development of the Calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces. A line with a chosen Cartesian system is called a number line.
Cartesian coordinate system
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The right hand rule.
Cartesian coordinate system
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate system
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3D Cartesian Coordinate Handedness
4.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures for dealing with lengths, areas, volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, curves, as well as the more advanced notions of manifolds and topology or metric. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense. The educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, analytic geometry. Euclidean geometry also has applications in computer science, various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry.
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
5.
Coordinate system
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The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems. Another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinate system
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The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
6.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, certain other spaces. It is named after the Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. Geometric shapes are defined as equations and inequalities. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. Euclidean spaces have finite dimension. One way to think of the Euclidean plane is as a set of points satisfying expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted by the same distance. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, so on. The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner.
Euclidean space
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A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Euclidean spaces
7.
Coordinate line
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The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the order of the coordinate axis the system may be a left-hand system. This is one of many coordinate systems. Another coordinate system for the plane is the coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinate line
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The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
8.
Cartesian coordinate
–
In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. The adjective Cartesian refers to the French Mathematician and Philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, the spherical and cylindrical coordinates for three-dimensional space. The development of the Cartesian coordinate system would play a fundamental role in the development of the Calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces. A line with a chosen Cartesian system is called a number line.
Cartesian coordinate
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The right hand rule.
Cartesian coordinate
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate
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3D Cartesian Coordinate Handedness
9.
Coordinate surfaces
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The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the order of the coordinate axis the system may be a left-hand system. This is one of many coordinate systems. Another coordinate system for the plane is the coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinate surfaces
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The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
10.
Cylindrical coordinate system
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The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The origin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The third coordinate may be called the height or altitude, longitudinal position, or axial position. They are sometimes called "cylindrical polar coordinates" and "polar cylindrical coordinates", are sometimes used to specify the position of stars in a galaxy. The three coordinates of a point P are defined as: The radial distance ρ is the Euclidean distance from the z-axis to the point P. The height z is the signed distance from the chosen plane to the point P. As in polar coordinates, the same point with cylindrical coordinates has infinitely many equivalent coordinates, namely and, where n is any integer. Moreover, if the radius ρ is zero, the azimuth is arbitrary. The notation for cylindrical coordinates is not uniform. The ISO standard 31-11 recommends, where ρ is the radial coordinate, φ the azimuth, z the height. In mathematical illustrations, a positive coordinate is measured counterclockwise as seen from any point with positive height. The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formulae may be used to convert between them. The function is assumed to return an angle in the =.
Cylindrical coordinate system
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A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4.
11.
Spherical coordinates
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It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the radius or radial coordinate. The polar angle may be called inclination angle. The use of symbols and the order of the coordinates differs between sources. In both systems ρ is often used instead of r. Other conventions are also used, so great care needs to be taken to check which one is being used. A number of different spherical coordinate systems following other conventions are used outside mathematics. In a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of coordinate systems based with different terms for the various coordinates. The polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon. The spherical system generalises the coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system. To define a spherical coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, an origin point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith.
Spherical coordinates
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Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
12.
Coordinate plane
–
The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems. Another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinate plane
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The right hand rule.
Coordinate plane
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Coordinate plane
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3D Cartesian Coordinate Handedness
13.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space, the surface of a completely round ball. The given point is the center of the mathematical ball. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the ball. The sphere share the same radius, diameter, center. The area of a sphere is: A = 4 π r 2. The total volume is the summation of all shell volumes: V ≈ ∑ A ⋅ r. In the limit as δr approaches zero this equation becomes: V = ∫ 0 r A d r ′. Substitute V: 4 3 π r 3 = ∫ 0 r A d r ′. Differentiating both sides of this equation with respect to r yields A as a function of r: 4 π r 2 = A. Which is generally abbreviated as: A = 4 π r 2. Alternatively, the element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. For more generality, see element. Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. The total volume is the summation of all incremental volumes: V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes: V = ∫ − r r π y 2 d x.
Sphere
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Circumscribed cylinder to a sphere
Sphere
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A two-dimensional perspective projection of a sphere
Sphere
Sphere
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Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres
14.
Scalar (mathematics)
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A scalar is an element of a field, used to define a vector space. A quantity described by multiple scalars, such as having both magnitude, is called a vector. More generally, a space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that space will be the elements of the associated field. A space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its part. The term is also sometimes used informally to mean a vector, matrix, other usually "compound" value, actually reduced to a single component. Thus, for example, the product of an n × 1 matrix, formally a 1 × 1 matrix, is often said to be a scalar. I is the identity matrix. The scalar derives from the Latin word scalaris, an adjectival form of scala. The English word "scale" also comes from scala. In a coordinate space, the scalar multiplication k yields. In a space, kƒ is the function x ↦ k. The scalars can be taken from any field, including complex numbers, as well as finite fields. According to a fundamental theorem of linear algebra, every space has a basis.
Scalar (mathematics)
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Scalars are real numbers used in linear algebra, as opposed to vectors. This image shows a Euclidean vector. Its coordinates x and y are scalars, as is its length, but v is not a scalar.
15.
Vector (geometric)
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In mathematics, physics, engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added according to algebra. A vector is what is needed to "carry" the A to the B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planet rotation around the Sun. The direction refers to B. Associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a space. Physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their direction can still be represented by the direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Vector-like objects that transform in a similar way under changes of the coordinate system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane and thus erected the first space of vectors in the plane.
Vector (geometric)
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This article is about the vectors mainly used in physics and engineering to represent directed quantities. For mathematical vectors in general, see Vector (mathematics and physics).
16.
Tensor
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Tensors are geometric objects that describe linear relations between geometric vectors, scalars, other tensors. Elementary examples of such relations include the dot product, linear maps. Geometric vectors, scalars themselves are also tensors. Given fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. For example, a linear map is represented by a matrix in a basis, therefore is a 2nd-order tensor. A vector is a 1st-order tensor. Scalars are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. The precise form of the law determines the type of the tensor. M is the number of covariant indices. The total order of a tensor is the sum of these two numbers. The concept enabled an alternative formulation of the intrinsic geometry of a manifold in the form of the Riemann curvature tensor. There are several approaches to defining tensors. Although seemingly different, the approaches just describe the geometric concept using different languages and at different levels of abstraction. For example, a linear operator is represented in a basis as a two-dimensional n × n array.
Tensor
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Cauchy stress tensor, a second-order tensor. The tensor's components, in a three-dimensional Cartesian coordinate system, form the matrix whose columns are the stresses (forces per unit area) acting on the e 1, e 2, and e 3 faces of the cube.
17.
Tensor analysis
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Tensors are geometric objects that describe linear relations between geometric vectors, scalars, other tensors. Elementary examples of such relations include the dot product, the cross product, linear maps. Geometric vectors, often used in physics and engineering applications, scalars themselves are also tensors. Given a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. For example, a linear map is represented by a matrix in a basis, therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. The precise form of the transformation law determines the type of the tensor. The tensor type is a pair of natural numbers, where n is the number of contravariant indices and m is the number of covariant indices. The total order of a tensor is the sum of these two numbers. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor. There are several approaches to defining tensors. Although seemingly different, the approaches just describe the same geometric concept using different languages and at different levels of abstraction. For example, a linear operator is represented in a basis as a two-dimensional square n × n array.
Tensor analysis
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Cauchy stress tensor, a second-order tensor. The tensor's components, in a three-dimensional Cartesian coordinate system, form the matrix whose columns are the stresses (forces per unit area) acting on the e 1, e 2, and e 3 faces of the cube.
18.
Gradient
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In mathematics, the gradient is a generalization of the usual concept of derivative to functions of several variables. If f is a real-valued function of several variables, its gradient is the vector whose components are the n partial derivatives of f. It is thus a vector-valued function. Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. The Jacobian is the generalization of the gradient for differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative. Consider a room in which the temperature is given by T, so at each point the temperature is T. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above level at point is H. The gradient of H at a point is a vector pointing in the direction of the steepest grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. Suppose that the steepest slope on a hill is 40%. If a road goes up the hill, then the steepest slope on the road will also be 40 %.
Gradient
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Gradient of the 2-d function f (x, y) = xe −(x 2 + y 2) is plotted as blue arrows over the pseudocolor plot of the function.
Gradient
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In the above two images, the values of the function are represented in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.
19.
Curl (mathematics)
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In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector characterize the rotation at that point. If the field represents the velocity of a moving fluid, then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. ∇ × F are often used for curl F. The connection is reflected in the ∇ × for the curl. As such, the curl operator maps continuously differentiable functions f: ℝ3 → ℝ3 to continuous functions g: ℝ3 → ℝ3. In fact, it maps Ck functions in ℝ3 to Ck − 1 functions in ℝ3. The above formula means that the curl of a field is defined as the infinitesimal density of the circulation of that field. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1. Suppose the field describes the field of a fluid flow and a small ball is located within the fluid or gas. If the ball has a rough surface, the fluid flowing past it will make it rotate. Such notation involving operators is common in physics and algebra.
Curl (mathematics)
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The components of F at position r, normal and tangent to a closed curve C in a plane, enclosing a planar vector area A = A n.
20.
Boundary conditions
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A solution to a boundary problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical equation will have them. Problems involving the equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important value problems are the Sturm -- Liouville problems. The analysis of these problems involves the eigenfunctions of a operator. To be useful in applications, a boundary problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions; the solution was given by the Dirichlet's principle. Boundary value problems are similar to initial value problems. Without the boundary conditions, the general solution to this equation is y = A + B cos. From the boundary condition y = 0 one obtains 0 = A ⋅ 0 + B ⋅ 1 which implies that B = 0. From the boundary y = 2 one finds 2 = A ⋅ 1 and so A = 2. One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is y = 2 sin . Itself is a Dirichlet boundary condition, or first-type boundary condition. A condition which specifies the value of the normal derivative of the function is a Neumann boundary condition, or second-type boundary condition.
Boundary conditions
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Shows a region where a differential equation is valid and the associated boundary values
21.
Earth sciences
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Earth science or geoscience is an all-embracing term for the fields of science related to the planet Earth. It can be considered to be a branch of planetary science, but with a much older history. There are both reductionist and holistic approaches to Earth sciences. Typically, Earth scientists use tools from geography, physics, chemistry, mathematics to build a quantitative understanding of how the Earth system works and evolves. Geology describes the rocky parts of the Earth's crust and its historic development. Major subdisciplines are petrology, geochemistry, engineering geology, sedimentology. Geophysics and geodesy investigate the shape of its reaction to its magnetic and gravity fields. Geophysicists explore the Earth's mantle well as the tectonic and seismic activity of the lithosphere. Geophysics is commonly used to supplement the work of geologists in developing a comprehensive understanding of crustal geology, particularly in exploration. See geophysical survey. Soil science covers the outermost layer of the Earth's crust, subject to soil formation processes. Major subdisciplines include edaphology and pedology. Ecology covers the interactions between the biota, with their natural environment. Hydrology describe the marine and freshwater domains of the watery parts of the Earth. Major subdisciplines include hydrogeology and physical, chemical, biological oceanography.
Earth sciences
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A volcanic eruption is the release of stored energy from below the surface of Earth, originating from radioactive decay and gravitational sorting in the Earth's core and mantle, and residual energy gained during the Earth's formation
Earth sciences
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The magnetosphere shields the surface of Earth from the charged particles of the solar wind. It is compressed on the day (Sun) side due to the force of the arriving particles, and extended on the night side. Image not to scale.
22.
Cartography
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Cartography is the study and practice of making maps. Combining science, technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively. The fundamental problems of traditional cartography are to: Set select traits of the object to be mapped. This is the concern of editing. Traits may be abstract, such as toponyms or political boundaries. Represent the terrain of the mapped object on flat media. This is the concern of map projections. Eliminate characteristics of the mapped object that are not relevant to the map's purpose. This is the concern of generalization. Reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization. Orchestrate the elements of the map to best convey its message to its audience. This is the concern of design. Modern cartography constitutes many practical foundations of geographic information systems. A painting, which may depict the ancient Anatolian city of Çatalhöyük, has been dated to the late 7th millennium BCE.
Cartography
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A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's Geography and using his second map projection. The translation into Latin and dissemination of Geography in Europe, in the beginning of the 15th century, marked the rebirth of scientific cartography, after more than a millennium of stagnation.
Cartography
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Valcamonica rock art (I), Paspardo r. 29, topographic composition, 4th millennium BC
Cartography
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The Bedolina Map and its tracing, 6th–4th century BC
Cartography
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Copy (1472) of St. Isidore's TO map of the world.
23.
Physics
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One of the main goal of physics is to understand how the universe behaves. Physics is one of perhaps the oldest through its inclusion of astronomy. The boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences while opening new avenues of research in areas such as philosophy. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. The United Nations named the World Year of Physics. Astronomy is the oldest of the natural sciences. The planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy. In the book, he was also the first to delved further into the way the eye itself works. Fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haytham's Optics ranks alongside that of Newton's work of the same title, published 700 years later. The translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the same devices as what Ibn Al Haytham understand the way light works. From this, important things as eyeglasses, magnifying glasses, telescopes, cameras were developed.
Physics
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Further information: Outline of physics
Physics
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Ancient Egyptian astronomy is evident in monuments like the ceiling of Senemut's tomb from the Eighteenth Dynasty of Egypt.
Physics
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Sir Isaac Newton (1643–1727), whose laws of motion and universal gravitation were major milestones in classical physics
Physics
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Albert Einstein (1879–1955), whose work on the photoelectric effect and the theory of relativity led to a revolution in 20th century physics
24.
Quantum mechanics
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Quantum mechanics, including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons. Systems such as these which obey quantum mechanics can be in a quantum superposition of different states, unlike in classical physics. Early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms. In one of them, the wave function, provides information about the probability amplitude of position, momentum, other physical properties of a particle. This experiment played a major role in the general acceptance of the theory of light. In 1838, Michael Faraday discovered cathode rays. Planck's hypothesis that energy is absorbed in discrete "quanta" precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Following Max Planck's solution to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld. This phase is known as old theory.
Quantum mechanics
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Max Planck is considered the father of the quantum theory.
Quantum mechanics
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Solution to Schrödinger's equation for the hydrogen atom at different energy levels. The brighter areas represent a higher probability of finding an electron
Quantum mechanics
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The 1927 Solvay Conference in Brussels.
25.
Theory of relativity
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The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity applies to their interactions, describing all their physical phenomena except gravity. General relativity explains the law of its relation to other forces of nature. It applies to the astrophysical realm, including astronomy. The theory transformed theoretical astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including spacetime as a unified entity of space and time, length contraction. In the field of physics, relativity improved the science of their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted astronomical phenomena such as neutron stars, black holes, gravitational waves. Max Planck, others did subsequent work. Einstein developed general relativity between 1907 and 1915, after 1915. The final form of general relativity was published in 1916. In the section of the same paper, Alfred Bucherer used for the first time the expression "theory of relativity". By the 1920s, the community understood and accepted special relativity. It rapidly became a necessary tool for theorists and experimentalists in the new fields of atomic physics, nuclear physics, quantum mechanics. By comparison, general relativity did not appear to be as useful, beyond making minor corrections to predictions of Newtonian theory.
Theory of relativity
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USSR stamp dedicated to Albert Einstein
Theory of relativity
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Key concepts
26.
Engineering
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The term Engineering is derived from ingeniare, meaning "to contrive, devise". Engineering has existed as humans devised fundamental inventions such as lever, wheel, pulley. Each of these inventions is essentially consistent with the modern definition of engineering. The engineering is derived from the engineer, which itself dates back to 1390 when an engine'er originally referred to "a constructor of military engines." In this context, now obsolete, an "engine" referred to a military machine, i.e. a mechanical contraption used in war. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, e.g. the U.S. Army Corps of Engineers. The word "engine" itself is of even older origin, ultimately deriving from the Latin ingenium, meaning "innate quality, especially mental power, hence a clever invention." The earliest civil engineer known by name is Imhotep. Ancient Greece developed machines in both civilian and military domains. The mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed. The first engine was built by Thomas Savery. The development of this device gave rise to the Industrial Revolution in the coming decades, allowing for the beginnings of mass production. Similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of the Scottish engineer James Watt gave rise to mechanical engineering.
Engineering
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The steam engine, a major driver in the Industrial Revolution, underscores the importance of engineering in modern history. This beam engine is on display in the Technical University of Madrid.
Engineering
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Relief map of the Citadel of Lille, designed in 1668 by Vauban, the foremost military engineer of his age.
Engineering
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The Ancient Romans built aqueducts to bring a steady supply of clean fresh water to cities and towns in the empire.
Engineering
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The International Space Station represents a modern engineering challenge from many disciplines.
27.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the dimension. In mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called Euclidean space. It is commonly represented by the ℝ3. This serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in this case, these three values can be labeled by any combination of three chosen from height, depth, breadth. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular at the origin, the point at which they cross. They are usually labeled x, y, z. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane.
Three-dimensional space
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Three-dimensional Cartesian coordinate system with the x -axis pointing towards the observer. (See diagram description for correction.)
28.
Position vector
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Usually denoted x, r, or s, it corresponds to the straight-line distances along each axis to P: r = O P →. The term "vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this can be easily generalized to Euclidean spaces in any number of dimensions. Corresponding basis vectors represent the same position vector. More general curvilinear coordinates are in contexts like continuum mechanics and general relativity. Linear algebra allows for the abstraction of an n-dimensional vector. The notion of "space" is intuitive since each xi can be any value, the collection of values defines a point in space. The dimension of the space is n. The coordinates of the r with respect to the basis vectors ei are xi. The vector of coordinates forms n-tuple. Each coordinate xi may be parameterized a number of parameters t. The linear span of a basis set B = equals the position R, denoted span = R. In the case of one dimension, the position has only one component, so it effectively degenerates to a coordinate. It could be, say, the radial r-direction. These derivatives have common utility in the study of kinematics, control theory, other sciences.
Position vector
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Space curve in 3D. The position vector r is parameterized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.
29.
Standard basis
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Here the vector ex points in the x direction, the ey points in the direction, the vector ez points in the z direction. There are several common notations for these vectors, including, and. These vectors are sometimes written with a hat to emphasize their status as unit vectors. Each of these vectors is sometimes referred to as the versor of the corresponding Cartesian axis. These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. Standard bases can be defined for other vector spaces, such as polynomials and matrices. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices M n, the standard basis consists of the m × n-matrices with exactly one non-zero entry, 1. For example, the standard basis for 2 × 2 matrices is formed by the 4 matrices e 12 =, e =, e 22 =. By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not necessarily a standard basis. There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials. The existence of other'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of theory called monomial theory.
Standard basis
–
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j, and k.
30.
Basis (linear algebra)
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In more general terms, a basis is a linearly independent spanning set. A B of a vector V over a field F is a linearly independent subset of V that spans V. In more detail, suppose that B = is a finite subset of a V over a field F. The numbers ai are called the coordinates of the vector x with respect to the basis B, by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. Settings that permit linear combinations allow alternative definitions of the concept: see Related notions below. Every vector in V can be expressed as a linear combination of vectors in B in a unique way. If the basis is ordered then the coefficients in this linear combination provide coordinates of the vector relative to the basis. Every vector space has a basis. The proof of this requires the axiom of choice. All bases of a space have the same cardinality, called the dimension of the space. This result is known as the theorem, requires a strictly weaker form of the axiom of choice. Also many vector sets can be attributed a standard basis which comprises both spanning and linearly independent vectors. Standard bases for example: In Rn, where ei is the column of the matrix.
Basis (linear algebra)
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This picture illustrates the standard basis in R 2. The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is linearly dependent upon them.
31.
Tangent
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In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. A similar definition applies in Euclidean space. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. The word "tangent" comes from the Latin tangere,'to touch'. Euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no straight line could fall between the curve. Archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself. These methods led to the development of differential calculus in the 17th century. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents. Further developments included those of Isaac Barrow, leading to the theory of Gottfried Leibniz. An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it". This old definition prevents inflection points from having any tangent.
Tangent
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Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
32.
Tangent bundle
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In differential geometry, the tangent bundle of a differentiable manifold M is a manifold T M, which assembles all the tangent vectors in M. As a set, it is given by the disjoint union of the tangent spaces of M. That is, T M = ⨆ x ∈ M T x M = ⋃ x ∈ M × T x M = ⋃ x ∈ M. Where T x M denotes the tangent space to M at the point x. There is a natural π: T M ↠ M defined by π = x. This projection maps each tangent space T M to the single point x. The bundle comes equipped with a natural topology. With this topology, the tangent bundle to a manifold is the prototypical example of a bundle. By definition, a manifold M is only if the tangent bundle is trivial. For example, the n-dimensional Sn is framed for all n, but parallelizable only for n = 1,3,7. One of the main roles of the tangent bundle is to provide a range for the derivative of a smooth function. Namely, if f: M → N is a smooth function, with M and N smooth manifolds, its derivative is a smooth Df: TM → TN. The bundle comes equipped with a natural topology and smooth structure so as to make it into a manifold in its own right. The dimension of TM is twice the dimension of M. Each tangent space of an n-dimensional manifold is an n-dimensional vector space.
Tangent bundle
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Informally, the tangent bundle of a manifold (in this case a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
33.
Fluid mechanics
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Fluid mechanics is a branch of physics concerned with the mechanics of fluids and the forces on them. Fluid mechanics has a wide range of applications, including for mechanical engineering, civil engineering, chemical engineering, geophysics, biology. Especially fluid dynamics, is an active field of research with many problems that are partly or wholly unsolved. Fluid mechanics can best be solved by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics, is devoted to this approach to solving fluid problems. An experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow. Viscous flow was explored by a multitude of engineers including Jean Léonard Marie Poiseuille and Gotthilf Hagen. Fluid hydrostatics is the branch of fluid mechanics that studies fluids at rest. Hydrostatics is fundamental to the engineering of equipment for storing, transporting and using fluids. It is also relevant to meteorology, to medicine, many other fields. Fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow -- the science of gases in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics. Some fluid-dynamical principles are used in traffic crowd dynamics. Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table. A fluid at rest has no stress.
Fluid mechanics
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Balance for some integrated fluid quantity in a control volume enclosed by a control surface.
34.
Continuum mechanics
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The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Research in the area continues today. Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Continuum mechanics deals with physical properties of fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience. Materials, such as solids, gases, are composed of molecules separated by "empty" space. On a microscopic scale, materials have discontinuities. A continuum is a body that can be continually sub-divided with properties being those of the bulk material. More specifically, the hypothesis/assumption hinges on the concepts of a representative elementary volume and separation of scales based on the Hill -- Mandel condition. The latter then provide a micromechanics basis for finite elements. The levels of SVE and RVE link continuum mechanics to statistical mechanics. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous. Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made. Consider traffic on a highway -- with just one lane for simplicity.
Continuum mechanics
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Figure 1. Configuration of a continuum body
35.
Differential geometry
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Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. Differential geometry developed to the mathematical analysis of surfaces. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address. Initially applied to the Euclidean space, further explorations led to non-Euclidean space, metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a symmetric bilinear form defined on the tangent space at each point. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.
Differential geometry
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A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
36.
Total differential
–
In calculus, the differential represents the principal part of the change in a function y = f with respect to changes in the independent variable. One also writes f = f ′ d x. The precise meaning of the variables dx depends on the context of the application and the required level of mathematical rigor. Traditionally, this interpretation is made rigorous in non-standard analysis. The dy/dx is not infinitely small; rather it is a real number. The use of infinitesimals in this form was widely criticized, for instance by The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential to the atomism of Leibniz's infinitesimals. In physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. Courant & John reconcile the physical use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent non-zero values that are smaller than the degree of accuracy required for the particular purpose for which they are intended. Thus "physical infinitesimals" need not appeal to a mathematical infinitesimal in order to have a precise sense. In real analysis, it is more desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a linear functional of an Δx. In non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing. The differential is defined in modern treatments of differential calculus as follows.
Total differential
–
The differential of a function ƒ (x) at a point x 0.
37.
Covariance and contravariance of vectors
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In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In physics, a basis is sometimes thought of as a set of reference axes. A change of scale on the reference axes corresponds to a change of units in the problem. In changing scale from meters to centimeters, the components of a measured vector will multiply by 100. Vectors exhibit this behavior of changing scale inversely to changes in scale to the reference axes: they are contravariant. As a result, vectors often have units of distance or distance times some other unit. In contrast, dual vectors typically have units the inverse of distance or the inverse of distance times some other unit. An example of a dual vector is the gradient, which has − 1. The components of dual vectors change in the same way as changes to scale of the reference axes: they are covariant. That is, the matrix that transforms the vector of components must be the inverse of the matrix that transforms the basis vectors. The components of vectors are said to be contravariant. In Einstein notation, contravariant components are denoted with upper indices as in v = v i e i. For a dual vector to be basis-independent, the components of the dual vector must co-vary with a change of basis to remain representing the same covector. That is, the components must be transformed by the same matrix as the change of basis matrix. The components of dual vectors are said to be covariant.
Covariance and contravariance of vectors
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tangent basis vectors (yellow, left: e 1, e 2, e 3) to the coordinate curves (black),
38.
Del
–
Del, or nabla, is an operator used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus. These formal products do not necessarily commute with other products. Del is used as a form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, Laplacian. The magnitude of the gradient is the value of this steepest slope. Note, a scalar. When operating on a vector it must be distributed to each component. The Laplacian is ubiquitous throughout mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, the Schrödinger equation. Del can also be applied with the result being a tensor. This quantity is equivalent to the transpose of the Jacobian matrix of the field with respect to space. The divergence of the field can then be expressed as the trace of this matrix. Because of the diversity of vector products one application of del already gives rise to three major derivatives: the gradient, curl. This is part of the value to be gained in notationally representing this operator as a vector. Whereas a vector is an object with both a direction, del has neither a magnitude nor a direction until it operates on a function.
Del
–
DCG chart: A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist.
39.
Linear operator
–
In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in which case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not. A linear map always maps linear subspaces onto linear subspaces; for instance it maps a plane to a plane, straight line or point. Simple examples include rotation and reflection linear transformations. In the language of abstract algebra, a linear map is a module homomorphism. In the language of theory it is a morphism in the category of modules over a given ring. Let V and W be vector spaces over the same field K. Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are considered as spaces over the K as above, we talk about K-linear maps. It is not C-linear. A linear map from V to K is called a linear functional. These statements generalize to any left-module RM over a ring R upon reversing of the scalar multiplication. The zero map between two left-modules over the same ring is always linear.
Linear operator
–
"Linear transformation" redirects here. For fractional linear transformations, see Möbius transformation.
40.
Covariant transformation
–
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. Conventionally, indices identifying the basis vectors so are all entities that transform in the same way. The inverse of a covariant transformation is a contravariant transformation. Since a vector should be invariant under a change of basis, its components must transform according to the contravariant rule. Conventionally, indices identifying the components of a vector so are all indices of entities that transform in the same way. The sum over matching indices of a product with the same lower and upper indices are invariant under a transformation. A vector itself is a geometrical quantity, in independent of the chosen basis. A v is given, say, in components vi on a chosen basis ei. A v is described in a given coordinate grid on a basis which are the tangent vectors to the coordinate grid. The basis vectors are ey. In another coordinate system, the new basis vectors are tangent vectors in the radial perpendicular to it. These basis vectors are indicated as er and eφ. They appear rotated anticlockwise to the first basis. The transformation here is thus an anticlockwise rotation.
Covariant transformation
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A vector v, and local tangent basis vectors { e x, e y } and { e r, e φ }.
41.
Origin (mathematics)
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In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect. The origin divides each of these axes into a negative semiaxis. The coordinates of the origin are all zero in three. In a polar coordinate system, the origin may also be called the pole. In Euclidean geometry, the origin may be chosen freely as any convenient point of reference. The origin of the complex plane can be referred as the point where real axis and imaginary axis intersect each other. In other words, it is the complex number zero.
Origin (mathematics)
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The origin of a Cartesian coordinate system
42.
Smooth function
–
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives in its domain. Differentiability class is a classification of functions according to the properties of their derivatives. Higher differentiability classes correspond to the existence of more derivatives. Consider an open set on a function f defined on that set with real values. Let k be a non-negative integer. The f is said to be of class Ck if the derivatives f ′, f ′ ′... f exist and are continuous. The f is said to be of class C ∞, or smooth, if it has derivatives of all orders. Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω. To put it differently, the C0 consists of all continuous functions. The C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative is of class C0. In particular, there are examples to show that this containment is strict. The class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers.
Smooth function
–
A bump function is a smooth function with compact support.
43.
Partial derivative
–
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in geometry. The partial-derivative symbol is ∂. One of the first known uses of the symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The derivative notation is by Adrien-Marie Legendre, though he later abandoned it; Carl Gustav Jacob Jacobi re-introduced the symbol in 1841. Suppose that ƒ is a function of more than one variable. For instance, z = f = x 2 + x y + y 2. The graph of this function defines a surface in Euclidean space. To every point on this surface, there is an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. Therefore ∂ z ∂ x = 3 at the point. That is, the partial derivative of z with respect to x at is 3, as shown in the graph. In other words, every value of y defines a function, denoted fy, a function of one variable x.
Partial derivative
–
A graph of z = x 2 + xy + y 2. For the partial derivative at (1, 1) that leaves y constant, the corresponding tangent line is parallel to the xz -plane.
44.
Systems of linear equations
–
In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. The word "system" indicates that the equations are to be considered collectively, rather than individually. Linear programming is a collection of methods for finding the "best" integer solution. Gröbner theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure. The simplest kind of linear system involves two variables: 2 x + 3 y = 6 4 x + 9 y = 15. One method for solving such a system is as follows. First, solve the top equation for x in terms of y: x = 3 − 2 y. Now substitute this expression for x into the bottom equation: 4 + 9 y = 15. This results in a single equation involving only the variable y. Solving gives y = 1, substituting this back into the equation for x yields x = 3 / 2. One extremely helpful view is that each unknown is a weight for a vector in a linear combination. X 1 + 2 + ⋯ + x n = This allows all the language and theory of vector spaces to be brought to bear. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique.
Systems of linear equations
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A linear system in three variables determines a collection of planes. The intersection point is the solution.
45.
Jacobian matrix
–
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature. Suppose f: ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f ∈ ℝm. This matrix, whose entries are functions of x, is also denoted by Df, Jf, ∂/∂. This linear map is thus the generalization of the usual notion of derivative, is called the derivative or the differential of f at x. If m = n, the Jacobian matrix is a square matrix, its determinant, a function of x1, …, xn, is the Jacobian determinant of f. It carries important information about the local behavior of f. The Jacobian determinant also appears when changing the variables in multiple integrals. These concepts are named after the mathematician Carl Gustav Jacob Jacobi. The Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a scalar-valued multivariate function is the gradient and that of a scalar-valued function of single variable is simply its derivative. The Jacobian can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that a transformation imposes locally. For example, if = f is used to transform an image, the Jacobian Jf, describes how the image in the neighborhood of is transformed. If p is a point in ℝn and f is differentiable at p, then its derivative is given by Jf. Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order: f = f + f ′ + o.
Jacobian matrix
–
A nonlinear map f: R 2 → R 2 sends a small square to a distorted parallelepiped close to the image of the square under the best linear approximation of f near the point.
46.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, subspaces, but is also concerned with properties common to all vector spaces. The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, initially listed as an advancement in geodesy. The study of algebra first emerged in the mid-1800s.
Linear algebra
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The three-dimensional Euclidean space R 3 is a vector space, and lines and planes passing through the origin are vector subspaces in R 3.
47.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced by Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the fraction 4/3, all the irrational numbers, such as √ 2. Included within the irrationals are the transcendental numbers, such as π. Complex numbers include real numbers. These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions are thus equivalent. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 19th centuries, there was much work on irrational and transcendental numbers. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, Ferdinand von Lindemann, showed that π is transcendental. Lindemann's proof has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly.
Real number
–
A symbol of the set of real numbers (ℝ)
48.
Set (mathematics)
–
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a well-defined collection of distinct objects. The objects that make up a set can be anything: other sets, so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description: A is the set whose members are the first four positive integers. B is the set of colors of the French flag. The second way is by extension –, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets: C = D =. One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D. There are two important points to note about sets.
Set (mathematics)
–
A set of polygons in a Venn diagram
49.
Real numbers
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced by Descartes, who distinguished between imaginary roots of polynomials. The real numbers include all the rational numbers, such as all the irrational numbers, such as √ 2. Included within the irrationals are the transcendental numbers, such as π. The real line can be thought of as a part of the complex plane, complex numbers include real numbers. These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the axiomatic definition and are thus equivalent. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 19th centuries, there was much work on transcendental numbers. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, Ferdinand von Lindemann, showed that π is transcendental. Lindemann's proof has finally been made elementary by Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly.
Real numbers
–
A symbol of the set of real numbers (ℝ)
50.
Cartesian product
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In mathematics, a Cartesian product is a mathematical operation that returns a set from multiple sets. Products can be specified using set-builder notation, e.g. A × B =. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. An ordered pair is a 2-tuple or couple. An illustrative example is the standard 52-card deck. The standard playing card ranks form a 13-element set. The card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards. Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct, even disjoint. The main historical example is the Cartesian plane in analytic geometry. Usually, such a pair's first and second components are called its x and y coordinates, respectively; cf. picture.
Cartesian product
–
Standard 52-card deck
Cartesian product
–
Cartesian product of the sets and
51.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. There are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a space. In the same vein, but in a more geometric sense, vectors representing displacements in three-dimensional space also form vector spaces. Infinite-dimensional vector spaces arise naturally as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of continuity. Among these topologies, those that are defined by inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Vector spaces are applied throughout mathematics, science and engineering. Furthermore, vector spaces furnish an coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading in geometry and abstract algebra. This is used in physics to describe velocities. Given any two such arrows, w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too.
Vector space
–
Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w.
52.
Coordinates
–
The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the order of the coordinate axis the system may be a left-hand system. This is one of many coordinate systems. Another coordinate system for the plane is the coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinates
–
The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
53.
Orthogonal vector
–
The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, engineering. The word comes from the Greek ὀρθός, meaning "upright", γωνία, meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y. The largest subspace of V, orthogonal to a given subspace is its orthogonal complement. Two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.
Orthogonal vector
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The line segments AB and CD are orthogonal to each other.
54.
Perpendicular
–
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects. A line is said to be perpendicular to another line if the two lines intersect at a right angle. For this reason, we may speak as being perpendicular without specifying an order. Perpendicularity easily extends to rays. In symbols, A B ¯ ⊥ C D ¯ means segment AB is perpendicular to line segment CD. A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines. Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of geometric objects. The word "foot" is frequently used with perpendiculars. This usage is exemplified in the top diagram, above, its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom. Step 2: construct circles centered at A' and B' having equal radius.
Perpendicular
–
The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
55.
Unit vector
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In mathematics, a unit vector in a normed vector space is a vector of length 1. A vector is often denoted by a lowercase letter with a circumflex, or "hat": ı ^. The term direction vector is used to describe a unit vector being used to represent spatial direction, such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are equivalent numerically to points on the circle. The same construct is used to specify spatial directions in 3D. As illustrated, each unique direction is equivalent numerically to a point on the sphere. The term normalized vector is sometimes used as a synonym for vector. Unit vectors are often chosen to form the basis of a space. Every vector in the space may be written as a linear combination of unit vectors. In a Euclidean space the dot product of two unit vectors is a scalar value amounting to the cosine of the smaller subtended angle. Unit vectors may be used to represent the axes of a coordinate system. They are often denoted using normal notation rather than standard unit vector notation. In most contexts it can be assumed that j, k, are versors of a 3-D Cartesian coordinate system. The notations, or, without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity.
Unit vector
–
Examples of two 2D direction vectors
56.
Scalar multiplication
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In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra. In geometrical contexts, multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar, must be distinguished from inner product of two vectors. The result of applying this function to c in K and v in V is denoted cv. Juxtaposition indicates the operation in the field. Scalar multiplication may be viewed as an action of the field on the space. A geometric interpretation of multiplication is that it stretches, or vectors by a constant factor. When V is Kn, multiplication may be defined as such. If K is not commutative, the distinct operations left right scalar vc may be defined. The left multiplication of a A with a scalar λ gives another matrix λA of the same size as A. The entries of λA are defined by j λ i j explicitly: λ A = λ =. When the underlying ring is commutative, for the real or complex field, these two multiplications are the same, are simply called scalar multiplication. However, for matrices over a more general ring that are not commutative, such as the quaternions, they may not be equal. For a real matrix: λ = 2, A = 2 = = = 2 = A 2.
Scalar multiplication
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Scalar multiplication of a vector by a factor of 3 stretches the vector out.
57.
Atlas (topology)
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In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as other fibre bundles. The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism φ to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair. An atlas for a topological space M is a collection of charts on M such that ⋃ U α = M. If the atlas is connected, then M is said to be an n-dimensional manifold. A map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. To be more precise, are two charts for a manifold M such that U α ∩ U β is non-empty. Note that since φ φ β are both homeomorphisms, the transition map τ α, β is also a homeomorphism. One often desires more structure than simply the topological structure.
Atlas (topology)
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Two charts on a manifold
58.
Differentiable manifold
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In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a structure locally by using the standard differential structure on a linear space. The maps that relate the coordinates defined by the various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable, smooth, holomorphic. A structure allows one to define the globally differentiable tangent space, differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as Yang -- Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry. The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen.
Differentiable manifold
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A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.
59.
Diffeomorphism
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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. If these functions are r times continuously differentiable, f is called a Cr-diffeomorphism. Two manifolds M and N are diffeomorphic if there is a diffeomorphism f from M to N. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. F is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. First remark It is essential for V to be simply connected for the function f to be globally invertible. This so-called Jacobian matrix is often used for explicit computations. Third remark Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective; and if n > k then Dfx could never be injective. In both cases, therefore, Dfx fails to be a bijection. Fourth remark If Dfx is a bijection at x then f is said to be a local diffeomorphism. Sixth remark A differentiable bijection is not necessarily a diffeomorphism. F = x3, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0.
Diffeomorphism
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Algebraic structure → Group theory Group theory
60.
Bijection
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In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other. A function maps elements to elements in its codomain. An injective function is an injection. Notationally: ∀ x, x ′ ∈ X, f = f ⇒ x = x ′. Or, equivalently, ∀ x, x ′ ∈ X, x ≠ x ′ ⇒ f ≠ f. The function is surjective if every element of the codomain is mapped to by at least one element of the domain. A surjective function is a surjection. Notationally: ∀ y ∈ Y, ∃ x ∈ X such that y = f. The function is bijective if every element of the codomain is mapped to by exactly one element of the domain. A bijective function is a bijection. A surjective function need not be injective. The four possible combinations of surjective features are illustrated in the diagrams to the right. A function is injective if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection.
Bijection
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Injective and surjective, i.e. bijective
61.
Domain of a function
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That is, the function provides value for each member of the domain. Given f: X → Y, the set X is the domain of f; the set Y is the codomain of f. In the expression f, f is the value. One can think of an argument as the value as the output. The image of f is the set of all values assumed by f for all possible x; this is the set. It can be a proper subset of it. It is, in general, smaller than the codomain; it is the whole codomain if and only if f is a surjective function. A well-defined function must map every element of its domain to an element of its codomain. For example, the function f defined by f = 1 / x has no value for f. Thus, the set of all real numbers, R, cannot be its domain. In cases like this, the function is either defined on R\ or the "gap is plugged" by explicitly defining f. Any function can be restricted to a subset of its domain. The restriction of g: A → B to S, where S ⊆ A, is written g |S: S → B. For instance the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain, the set of possible values of the function is typically called its range.
Domain of a function
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Illustration showing f, a function from pink domain X to blue and yellow codomain Y. The smaller yellow oval inside Y is the image of f. Either the image or the codomain also sometimes is called the range of f.
62.
Jacobian matrix and determinant
–
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature. Suppose f: ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f ∈ ℝm. This matrix, whose entries are functions of x, is also denoted by Df, Jf, ∂/∂. This linear map is thus the generalization of the usual notion of derivative, is called the derivative or the differential of f at x. If m = n, the Jacobian matrix is a square matrix, its determinant, a function of x1, …, xn, is the Jacobian determinant of f. It carries important information about the local behavior of f. The Jacobian determinant also appears when changing the variables in multiple integrals. These concepts are named after the mathematician Carl Gustav Jacob Jacobi. The Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a scalar-valued multivariate function is the gradient and that of a scalar-valued function of single variable is simply its derivative. The Jacobian can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that a transformation imposes locally. For example, if = f is used to transform an image, the Jacobian Jf, describes how the image in the neighborhood of is transformed. If p is a point in ℝn and f is differentiable at p, then its derivative is given by Jf. Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order: f = f + f ′ + o.
Jacobian matrix and determinant
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A nonlinear map f: R 2 → R 2 sends a small square to a distorted parallelepiped close to the image of the square under the best linear approximation of f near the point.
63.
Mechanics
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The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes. During the modern period, scientists such as Khayaam, Galileo, Kepler, Newton, laid the foundation for what is now known as classical mechanics. It can also be defined as a branch of science which deals with forces on objects. Historically, classical mechanics came first, while quantum mechanics is a comparatively recent invention. Classical mechanics originated in Principal Mathematical; Quantum Mechanics was discovered in the early 20th century. Both are commonly held to constitute the most certain knowledge that exists about physical nature. Classical mechanics has often been viewed as a model for other so-called exact sciences. Essential in this respect is the relentless use of mathematics in theories, well as the decisive role played by experiment in generating and testing them. Quantum mechanics is of a bigger scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers. However, for macroscopic processes classical mechanics is well used. Modern descriptions of such behavior begin as displacement, time, velocity, acceleration, mass, force. Until about 400 years ago, however, motion was explained from a very different point of view. He showed that the speed of falling objects increases steadily during the time of their fall.
Mechanics
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Arabic Machine Manuscript. Unknown date (at a guess: 16th to 19th centuries).
64.
Tensor product
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The space is thus the "freest" such vector space, in the sense of having the least constraints. In each such case the product is characterized by a universal property: it is the freest bilinear operation. The ⊠ variant of ⊗ is used in control theory. The product of W over a field K is another vector space over K. It is denoted V ⊗ W when the underlying K is understood. This product operation ⊗: V × W → V ⊗ W is quickly verified to be bilinear. The disadvantage of the above definition is that it involves a choice of basis, which can not be done canonically for a generic vector space. However, it is not difficult to show that any two choices of basis lead to isomorphic tensor product spaces. Alternatively, the product may be defined in an basis-independent manner as a quotient space of a free vector space over V × W. This approach is described below. The definition of ⊗ requires the notion of the free F on some set S, a vector space whose basis is parameterized by S. It is a space with the usual addition and multiplication of functions. Because of this explicit expression, an element of F is often called a formal sum of symbols in S. By construction, the dimension of the vector space F equals the cardinality of the set S. Let us first consider a special case: let us say V, W are free vector spaces for the sets S, T respectively.
Tensor product
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This commutative diagram presents the universal property of tensor product.
65.
Metric (mathematics)
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In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. Not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable. A metric tensor thus determines a metric. However, not every metric comes from a metric tensor in this way. The first condition is implied by the others. These conditions express intuitive notions about the concept of distance. The inequality means that the distance from x to z via y is at least as great as from x to z directly. Euclid in his work stated that the shortest distance between two points is a line;, the inequality for his geometry. If a modification of the triangle inequality 4*. D ≤ d + d is used in the definition then property 1 follows straight from property 4*. 4 * give property 3 which in turn gives property 4. The discrete metric: if x = y then d = 0. Otherwise, d = 1.
Metric (mathematics)
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An illustration comparing the taxicab metric versus the Euclidean metric on the plane: In the taxicab metric all three pictured paths (red, yellow, and blue) have the same length (12) for the same route. In the Euclidean metric, the green path has length, and is the unique shortest path.
66.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. The concept of space is considered to be to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, part of a conceptual framework. Many of these philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute -- in the sense that it existed independently of whether there was any matter in the space. Kant referred to the experience of "space" as being a subjective "pure a priori form of intuition". In the 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. In the seventeenth century, the philosophy of time emerged as a central issue in epistemology and metaphysics. At its heart, the English physicist-mathematician, set out two opposing theories of what space is. Unoccupied regions are those that could have objects in them, thus spatial relations with other places. Space could be thought in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people. According to the principle of sufficient reason, any theory of space that implied that there could be these two possible universes must therefore be wrong.
Space
–
Gottfried Leibniz
Space
–
A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space.
Space
–
Isaac Newton
Space
–
Immanuel Kant
67.
Orthogonal
–
The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, engineering. The word comes from the Greek ὀρθός, meaning "upright", γωνία, meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y. The largest subspace of V, orthogonal to a given subspace is its orthogonal complement. Two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.
Orthogonal
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The line segments AB and CD are orthogonal to each other.
68.
Levi-Civita symbol
–
It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. There are nn indexed values of εi1i2…in, which can be arranged into an n-dimensional array. The key definitive property of the symbol is total antisymmetry in all the indices. If any two indices are equal, the symbol is zero. The value ε12…n must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε12…n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. The three- and higher-dimensional Levi-Civita symbols are used more commonly. In three dimensions only, the cyclic permutations of are all even permutations, similarly the anticyclic permutations are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of and easily obtain all the even or odd permutations. The formula is valid for all index values, for any n. A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol is sometimes called a permutation tensor.
Levi-Civita symbol
–
For the indices (i, j, k) in ε ijk, the values 1, 2, 3 occurring in the cyclic order (1,2,3) (yellow) correspond to ε = +1, while occurring in the reverse cyclic order (red) correspond to ε = −1, otherwise ε = 0.
69.
Dot product
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In mathematics, the dot product or scalar product, is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the cosine of the angle between them. In three-dimensional space, the product contrasts with the cross product of two vectors, which produces a pseudovector as the result. The product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions. The product may be defined algebraically or geometrically. The geometric definition is based on the notions of distance. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In such a presentation, the notions of length and angles are not primitive. For instance, in three-dimensional space, the product of vectors and is: ⋅ = + + = 4 − 6 + 5 = 3. In Euclidean space, a Euclidean vector is a geometrical object that possesses both a direction. A vector can be pictured as an arrow. Its direction is the direction that the arrow points. The magnitude of a vector a is denoted by ∥ a ∥.
Dot product
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Scalar projection
70.
Cross product
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It has many applications in mathematics, physics, engineering, programming. It should not be confused with product. If two vectors have the same direction or if either one has zero length, then their cross product is zero. The cross product is distributive over addition. But if the product is limited with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, then only the cross product qualifies. The cross product of two vectors b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes a ∧ b is used, though this is avoided in mathematics to avoid confusion with the exterior product. If b are parallel, by the above formula, the cross product of a and b is the zero vector 0. Then, the n is coming out of the thumb. Using this rule implies that the cross-product is anti-commutative, i.e. b × a = −. Using the cross product requires the handedness of the coordinate system to be taken into account. If a left-handed system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction. This, however, creates a problem because transforming to another should not change the direction of n. The problem is clarified by realizing that the cross product of two vectors is not a vector, but rather a pseudovector.
Cross product
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The cross-product in respect to a right-handed coordinate system
71.
Permutation symbol
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It is named after physicist Tullio Levi-Civita. Other names include the permutation symbol, alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are ϵ, or less commonly the Latin lower case e. There are nn indexed values of εi1i2…in, which can be arranged into an n-dimensional array. The definitive property of the symbol is total antisymmetry in all the indices. If any two indices are equal, the symbol is zero. The ε12... n must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε12…n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article. The values of the Levi-Civita symbol coordinate system. The three - and Levi-Civita symbols are used more commonly. In three dimensions only, the cyclic permutations of are all even permutations, similarly the anticyclic permutations are all permutations. This means in 3d it is sufficient to easily obtain all the even or odd permutations. The formula is valid for any n. A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol is sometimes called a tensor.
Permutation symbol
–
For the indices (i, j, k) in ε ijk, the values 1, 2, 3 occurring in the cyclic order (1,2,3) (yellow) correspond to ε = +1, while occurring in the reverse cyclic order (red) correspond to ε = −1, otherwise ε = 0.
72.
Curvilinear coordinates
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In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation, locally invertible at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The coordinates, coined by the French Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of coordinate systems in Euclidean space are Cartesian, spherical polar coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere, curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Curvilinear coordinates are often used to define the distribution of physical quantities which may be, for example, tensors. Such expressions then become valid for any curvilinear coordinate system. Depending on the application, a system may be simpler to use than the coordinate system. For instance, a physical problem with spherical symmetry defined in R3 is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as engineering. For now, consider 3d space.
Curvilinear coordinates
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Curvilinear, affine, and Cartesian coordinates in two-dimensional space
73.
Surface integral
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In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields, vector fields. Surface integrals have applications in physics, particularly with the theories of classical electromagnetism. Let such a parameterization be x, where varies in some T in the plane. So that ∂ r ∂ ∂ r ∂ y =. One can recognize the vector in the second line above as the normal vector to the surface. Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space. Consider a vector v on S, for each x in S, v is a vector. The integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. Alternatively, if we integrate the normal component of the field, the result is a scalar. Imagine that we have a fluid flowing through S, such that v determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S per time. The cross product on the right-hand side of this expression is a surface normal determined by the parametrization. This formula defines the integral on the left.
Surface integral
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The definition of surface integral relies on splitting the surface into small surface elements.
74.
Volume integral
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In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, it is a special case of multiple integrals. Volume integrals are especially important for example, to calculate flux densities. A volume integral is far more powerful. "Volume integral". MathWorld.
Volume integral
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Volume element in spherical coordinates
75.
Integration (mathematics)
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with differentiation, being the other. The area above the x-axis adds below the x-axis subtracts from the total. The operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is written: F = ∫ f d x. The integrals discussed in this article are those termed definite integrals. A mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a region by breaking the region into thin vertical slabs. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. A similar method was independently developed by Liu Hui, who used it to find the area of the circle. This method was later used by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. The significant advances in integral calculus did not begin to appear until the 17th century. Further steps were made in the 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus.
Integration (mathematics)
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A definite integral of a function can be represented as the signed area of the region bounded by its graph.
76.
Albert Einstein
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Albert Einstein was a German-born theoretical physicist. Einstein developed the general theory of one of the two pillars of modern physics. Einstein's work is also known on the philosophy of science. Einstein is best known in popular culture for his mass -- energy equivalence E = mc2. This led him to develop his special theory of relativity. Einstein continued to deal with problems of statistical mechanics and theory, which led to his explanations of particle theory and the motion of molecules. Einstein also investigated the thermal properties of light which laid the foundation of the theory of light. In 1917, he applied the general theory of relativity to model the large-scale structure of the universe. Einstein settled in the U.S. becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project. He largely denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, he signed the Russell -- Einstein Manifesto, which highlighted the danger of nuclear weapons. He was affiliated with the Institute until his death in 1955. He published more than 300 scientific papers along over 150 non-scientific works. On 5 universities and archives announced the release of Einstein's papers, comprising more than 30,000 unique documents.
Albert Einstein
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Albert Einstein in 1921
Albert Einstein
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Einstein at the age of 3 in 1882
Albert Einstein
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Albert Einstein in 1893 (age 14)
Albert Einstein
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Einstein's matriculation certificate at the age of 17, showing his final grades from the Argovian cantonal school (Aargauische Kantonsschule, on a scale of 1–6, with 6 being the highest possible mark)
77.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not. Manifolds naturally arise as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate.
Manifold
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The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Manifold
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The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
78.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. In particular, the curvature of spacetime is directly related to the momentum of whatever matter and radiation are present. The relation is specified by a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational time delay. The predictions of general relativity have been confirmed to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. Einstein's theory has astrophysical implications. General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics LIGO. In addition, general relativity is the basis of cosmological models of a consistently expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his relativistic framework. The Einstein field equations are very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. The objects known today as black holes. In 1917, Einstein applied his theory as a whole initiating the field of relativistic cosmology.
General relativity
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A simulated black hole of 10 solar masses within the Milky Way, seen from a distance of 600 kilometers.
General relativity
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Albert Einstein developed the theories of special and general relativity. Picture from 1921.
General relativity
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Einstein cross: four images of the same astronomical object, produced by a gravitational lens
General relativity
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Artist's impression of the space-borne gravitational wave detector LISA
79.
Solid mechanics
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Solid mechanics is fundamental for civil, aerospace, nuclear, mechanical engineering, for many branches of physics such as materials science. It has specific applications such as understanding the anatomy of living beings, the design of dental prostheses and surgical implants. One of the most common practical applications of solid mechanics is the Euler-Bernoulli equation. Solid mechanics extensively uses tensors to describe stresses, the relationship between them. As shown in the following table, solid mechanics inhabits a central place within continuum mechanics. The field of rheology presents an overlap between fluid mechanics. Its shape departs away from the rest shape due to stress. The amount of departure from shape is called deformation, the proportion of deformation to original size is called strain. This region of deformation is known as the linearly elastic region. It is most common for analysts in solid mechanics to use linear material models, due to ease of computation. However, real materials often exhibit non-linear behavior. As old ones are pushed to their limits, non-linear material models are becoming more common. Those that deform proportionally to the applied load, can be described by the linear elasticity equations such as Hooke's law. This implies that the response has time-dependence. Plastically – Materials that behave elastically generally do so when the applied stress is less than a yield value.
Solid mechanics
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Continuum mechanics
80.
Plate theory
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In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined compared to the planar dimensions. The typical thickness to ratio of a plate structure is less than 0.1. A theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of theory is to calculate the deformation and stresses in a plate subjected to loads. Of the numerous plate theories that have been developed since the 19th century, two are widely accepted and used in engineering. These are the Kirchhoff -- The Mindlin -- Reissner theory of plates Note: the Einstein summation convention of summing on repeated indices is used below. The Kirchhoff -- theory is an extension of Euler -- Bernoulli beam theory to thin plates. The theory was developed by Love using assumptions proposed by Kirchhoff. It is assumed that a mid-surface plane can be used to represent the three-dimensional plate in two-dimensional form. If α are the angles of rotation of the normal to the mid-surface, then in the Kirchhoff -- Love theory φ α = w, α 0. The equilibrium equations for the plate can be derived from the principle of virtual work. The quantities σ β are the stresses. It is more convenient to work with the moment results that enter the equilibrium equations. These are related by = and = −.
Plate theory
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Vibration mode of a clamped square plate
81.
Maxwell's equations
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One important consequence of the equations is that fluctuating electric and magnetic fields can propagate at the speed of light. This electromagnetic radiation manifests itself in manifold ways from radio waves to light and X- or γ-rays. The equations have two major variants. The microscopic Maxwell equations have universal applicability but may be infeasible to calculate with. They relate the magnetic fields including the complicated currents in materials at the atomic scale. The "macroscopic" Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. However, their use requires experimentally determining parameters for a phenomenological description of the electromagnetic response of materials. The term "Maxwell's equations" is often used for equivalent alternative formulations. The space-time formulations, are commonly used in high energy and gravitational physics because they make the compatibility of the equations with special and general relativity manifest. In many situations, though, deviations from Maxwell's equations are immeasurably small. Exceptions include many other phenomena related to photons or virtual photons. In the electric and magnetic field formulation there are four equations. The two inhomogeneous equations describe how the fields vary in space due to sources. Gauss's law describes how electric fields emanate from electric charges. Gauss's law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles.
Maxwell's equations
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Maxwell's equations (mid-left) as featured on a monument in front of Warsaw University's Centre of New Technologies
Maxwell's equations
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Electromagnetism
Maxwell's equations
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In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical power grids. Artist's rendition; sizes are not to scale.
Maxwell's equations
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Magnetic core memory (1954) is an application of Ampère's law. Each core stores one bit of data.
82.
Metamaterials
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A metamaterial is a material engineered to have a property, not found in nature. They are made from assemblies of multiple elements fashioned from composite materials such as metals or plastics. The materials are usually arranged in repeating patterns, at scales that are smaller than the wavelengths of the phenomena they influence. Metamaterials derive their properties not from the properties of the base materials, but from their newly designed structures. Appropriately designed metamaterials can sound in bulk materials. Those that exhibit a negative index of refraction for particular wavelengths have attracted significant research. These materials are known as negative-index metamaterials. Metamaterials offer the potential to create superlenses. Such a lens could allow imaging below the diffraction limit, the minimum resolution that can be achieved by conventional glass lenses. A form of'invisibility' was demonstrated using gradient-index materials. Acoustic and seismic metamaterials are also research areas. Explorations of artificial materials for manipulating electromagnetic waves began at the end of the 19th century. Some of the earliest structures that may be considered metamaterials were studied by Jagadish Chandra Bose, who in 1898 researched substances with chiral properties. Karl Ferdinand Lindman studied interaction as artificial chiral media in the early twentieth century. Winston E. Kock developed materials that had similar characteristics to metamaterials in the late 1940s.
Metamaterials
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Negative index metamaterial array configuration, which was constructed of copper split-ring resonators and wires mounted on interlocking sheets of fiberglass circuit board. The total array consists of 3 by 20×20 unit cells with overall dimensions of 10×100×100 mm.
83.
Chain rule
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In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. This can be written more explicitly in terms of the variable. Equivalently, F = f for all x. Then one can also write F ′ = ′ g ′. The rule may be written, in Leibniz's notation, in the following way. The rule then states, d z d x = d z d y ⋅ d y d x. In integration, the counterpart to the rule is the substitution rule. The rule seems to have first been used by Leibniz. He first mentioned it in a 1676 memoir. The common notation of rule is due to Leibniz. L'Hôpital uses the rule implicitly in his Analyse des infiniment petits. The rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery. Suppose that a skydiver jumps from an aircraft. Assume that t seconds after his height above sea level in meters is given by g = 4000 − 4.9 t2. One model for the atmospheric pressure at a h is f = 101325 e − 0.0001 h.
Chain rule
84.
Tangent plane
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The elements of the tangent space are called tangent vectors at x. This is a generalization of the notion of a bound vector in a Euclidean space. All the tangent spaces of a connected manifold have the same dimension, equal to the dimension of the manifold. More generally, if a given manifold is thought as an embedded submanifold of Euclidean space one can picture a tangent space in this literal fashion. Used by Dirac. More strictly this defines an affine space, distinct from the space of tangent vectors described by modern terminology. The points P at which the dimension is exactly that of V are called the non-singular points; the others are singular points. For example, a curve that crosses itself doesn't have a unique line at that point. The singular points of V are those where the'test to be a manifold' fails. See Zariski tangent space. Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the field of particles moving on a manifold. A field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. However, it is more convenient to define the notion of tangent space based on the manifold itself. There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via velocities of curves is intuitively the simplest, it is also the most cumbersome to work with.
Tangent plane
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A pictorial representation of the tangent space of a single point, x, on a sphere. A vector in this tangent space can represent a possible velocity at x. After moving in that direction to another nearby point, one's velocity would then be given by a vector in the tangent space of that nearby point—a different tangent space, not shown.
85.
Tensor derivative (continuum mechanics)
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The derivatives of scalars, vectors, second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the design of algorithms for numerical simulations. The directional derivative provides a systematic way of finding these derivatives. The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken. Let f be a real valued function of the vector v. Note: the Einstein summation convention of summing on repeated indices is used below. If T is a field of order n > 1 then the divergence of the field is a tensor of order n − 1. Note: the Einstein summation convention of summing on repeated indices is used below. In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field S. Note: the Einstein summation convention of summing on repeated indices is used below. Consider an arbitrary constant vector c. In an orthonormal basis, the components of A can be written as a A. In that case, the right side corresponds the cofactors of the matrix. Let A be a second tensor.
Tensor derivative (continuum mechanics)
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Domain, its boundary and the outward unit normal
86.
Centrifugal force
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The term has sometimes also been used for the force, a reaction to a centripetal force. All measurements of position and velocity must be made relative to some frame of reference. An inertial frame of reference is one, not accelerating. In terms of an inertial frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newton's laws of motion and the real forces. In its current usage the term'centrifugal force' has no meaning in an inertial frame. In an inertial frame, an object that has no forces acting on it travels in a straight line, according to Newton's first law. If it is desired to apply Newton's laws in the rotating frame, it is necessary to introduce new, fictitious, forces to account for this curved motion. This is the centrifugal force. Consider a stone being whirled round on a string, in a horizontal plane. The only real force acting on the stone in the horizontal plane is the tension in the string. There are no other forces acting on the stone so there is a net force on the stone in the horizontal plane. In order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed the stone moves in a straight line. In a frame of reference rotating with the stone around the same axis as the stone, the stone is stationary.
Centrifugal force
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The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
87.
Curvilinear perspective
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Curvilinear perspective is a graphical projection used to draw 3D objects on 2D surfaces. In 1959, Flocon had acquired a copy of Grafiek en tekeningen by M. C. Escher who strongly impressed him with his use of bent and curved perspective, which influenced the theory Flocon and Barre were developing. They started a long correspondence, in which Escher called Flocon a "kindred spirit". This technique can, like two-point perspective, use a vertical line as a line, creating both a worms and birds eye view at the same time. Earlier, less mathematically precise versions can be seen in the work of the miniaturist Jean Fouquet. Leonardo da Vinci in a lost notebook spoke of curved perspective lines. Examples of five-point perspective can also be found in the self-portrait of the mannerist painter Parmigianino seen through a shaving mirror. Another example would be the curved mirror in Arnolfini's Wedding by the Flemish painter Jan van Eyck. The book Vanishing Point: Perspective for Comics Up by Jason Cheeseman-Meyer teaches five and four point perspective. The ellipse has the property that its long axis is a diameter of the "bounding circle". Graphical projection Perspective projection distortion linear perspective Mathematics and art M. C. Escher Curvilinear coordinates Drawing Comics - 5-Point Perspective House of Stairs by M. C. Escher
Curvilinear perspective
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Curvilinearity in photography: Curvilinear (above) and rectilinear (below) image. Notice the barrel distortion typical for fisheye lenses in the curvilinear image. While this example has been rectilinear-corrected by software, high quality wide-angle lenses are built with optical rectilinear correction.
Curvilinear perspective
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Curvilinear barrel distortion
Curvilinear perspective
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Jean Fouquet, Arrival of Emperor Charles IV at the Basilica St Denis
Curvilinear perspective
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Parmigianino, Self-portrait in a Convex Mirror
88.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
89.
PubMed Identifier
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PubMed is a free search engine accessing primarily the MEDLINE database of references and abstracts on life sciences and biomedical topics. The United States National Library of Medicine at the National Institutes of Health maintains the database as part of the Entrez system of information retrieval. From 1971 to 1997, MEDLINE online access to the MEDLARS Online computerized database had been primarily through institutional facilities, such as university libraries. PubMed, first released in January 1996, ushered in the era of private, free, home- and office-based MEDLINE searching. Information about the journals indexed in MEDLINE and available through PubMed is found in the NLM Catalog. As of the same date, 13.1 million of PubMed's records are listed with their abstracts, 14.2 million articles have links to full-text. In 2016, NLM changed the indexing system so that publishers will be able to directly correct typos and errors in PubMed indexed articles. Simple searches on PubMed can be carried out by entering key aspects of a subject into PubMed's search window. When a journal article is indexed, numerous article parameters are extracted and stored as structured information. Such parameters are: Article Type, publication history. Publication type parameter enables many special features. Since July 2005, the MEDLINE process puts those in a field called Secondary Identifier. The secondary field is to store accession numbers to various databases of clinical trial IDs. For clinical trials, PubMed extracts trial IDs for the two largest trial registries: ClinicalTrials.gov and the International Standard Randomized Controlled Trial Number Register. A reference, judged particularly relevant can be marked and "related articles" can be identified.
PubMed Identifier
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PubMed
90.
Wikiversity
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Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, resulting activities. Wikiversity's beta phase officially began on August 15, 2006, with the English language Wikiversity. Two proposals were made. The first project proposal was not approved and the second, modified proposal, was approved. The launch of Wikiversity was announced at Wikimania 2006 as:.... We're also going to be hosting and fostering research into how these kinds of things can be used more effectively. Wikiversity is a center for the creation of and use of free learning materials, the provision of learning activities. Wikiversity is one of many wikis used in educational contexts, as well as many initiatives that are creating free and open educational resources. The primary goals for Wikiversity are to: host a range of multilingual learning materials/resources, for all age groups in all languages. Host scholarly/learning projects and communities that support these materials. The Wikiversity e-Learning model places emphasis on "learning groups" and "learning by doing". Wikiversity's motto and slogan is "set learning free", indicating that groups/communities of Wikiversity participants will engage in learning projects. Learning is facilitated on projects that are detailed, results reported by editing Wikiversity pages. Wikiversity learning projects include collections of wiki webpages concerned with the exploration of a particular topic. Wikiversity participants are encouraged to express their learning goals, the Wikiversity community collaborates to develop learning activities and projects to accommodate those goals.
Wikiversity
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Wikiversity
91.
Orthogonal coordinate system
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In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles. A coordinate surface for a coordinate qk is the hypersurface on which qk is a constant. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. The chief advantage of non-Cartesian coordinates is that they can be chosen to match the symmetry of the problem. The reason to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity: many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates many problems may be solved by separation of variables. Separation of variables is a mathematical technique that converts a complex d-dimensional problem into d one-dimensional problems that can be solved in terms of known functions. Many equations can be reduced to Laplace's equation or the Helmholtz equation. Laplace's equation is separable in 13 orthogonal coordinate systems, the Helmholtz equation is separable in 11 orthogonal coordinate systems. Orthogonal coordinates never have off-diagonal terms in their metric tensor. These scaling functions hi are used to calculate differential operators in the new coordinates, e.g. the gradient, the Laplacian, the divergence and the curl. A simple method for generating orthogonal coordinates systems in two dimensions is by a conformal mapping of a standard two-dimensional grid of Cartesian coordinates. A complex number z = x + iy can be formed from the real coordinates x and y, where i represents the square root of -1. However, there are coordinate systems in three dimensions that can not be obtained by rotating a two-dimensional system, such as the ellipsoidal coordinates. More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering their orthogonal trajectories.
Orthogonal coordinate system
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A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained.
92.
Polar coordinate system
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The reference point is called the pole, the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the angle is called the angular coordinate, polar angle, or azimuth. The concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis.
Polar coordinate system
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Hipparchus
Polar coordinate system
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Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
Polar coordinate system
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A planimeter, which mechanically computes polar integrals
93.
Parabolic coordinates
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Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g. the potential theory of the edges. The foci of all these parabolae are located at the origin. The two-dimensional parabolic coordinates form the basis for two sets of orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the z -direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. The foci of all these paraboloids are located at the origin. Parabolic cylindrical coordinates Orthogonal coordinate system Curvilinear coordinates Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. P. 660. ISBN 0-07-043316-X. LCCN 52011515. Margenau H, Murphy GM.
Parabolic coordinates
–
Contents
94.
Bipolar coordinates
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Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. The first is based on the Apollonian circles. The curves of constant σ and of τ are circles that intersect at right angles. The coordinates have two foci F2, which are generally taken to be fixed at and, respectively, on the x-axis of a coordinate system. The second system is two-center bipolar coordinates. There is also a third coordinate system, based on two poles. The term "bipolar" is sometimes used to describe other curves having two singular points, such as Cassini ovals. The centers of the constant-σ circles lie on the y-axis. Circles of positive σ are centered above the x-axis, whereas those of negative σ lie below the axis. The centers of the constant-τ circles lie on the x-axis. The τ = 0 curve corresponds to the y-axis. As the magnitude of τ increases, the radius of the circles decreases and their centers approach the foci. A typical example would be the electric field surrounding two parallel cylindrical conductors. Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates.
Bipolar coordinates
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Bipolar coordinate system
95.
Elliptic coordinates
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In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. On the complex plane, an relationship is x + i y = a cosh These definitions correspond to ellipses and hyperbolae. In an coordinate system the lengths of the basis vectors are known as scale factors. An geometrically intuitive set of elliptic coordinates are sometimes used, where σ = cosh μ and τ = cos ν. Hence, the curves of constant σ are ellipses, whereas the curves of constant τ are hyperbolae. The coordinate τ must belong to the interval, whereas the coordinate must be greater than or equal to one. The coordinates have a simple relation to the distances to F 2. Thus, the distance to F 1 is a, whereas the distance to F 2 is a. X = a σ τ y 2 = a 2. Elliptic coordinates form the basis for several sets of orthogonal coordinates. The elliptic cylindrical coordinates are produced by projecting in the z -direction. Some traditional examples are solving systems such as electrons orbiting a planetary orbits that have an elliptical shape. The geometric properties of elliptic coordinates can also be useful. Curvilinear coordinates Generalized coordinates Hazewinkel, Michiel, ed. "Elliptic coordinates", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Korn GA and Korn TM. Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
Elliptic coordinates
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Elliptic coordinate system
96.
Spherical coordinate system
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It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the radius or radial coordinate. The polar angle may be called angle. The use of symbols and the order of the coordinates differs between sources. In both systems ρ is often used instead of r. Other conventions are also used, so great care needs to be taken to check which one is being used. A number of different spherical coordinate systems following other conventions are used outside mathematics. In a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of different celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon. The spherical coordinate system generalises the two-dimensional polar coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system. To define a spherical coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, an origin point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith.
Spherical coordinate system
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Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
97.
Parabolic cylindrical coordinates
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Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g. the potential theory of edges. The foci of all these parabolic cylinders are located along the line defined by x = y = 0. A typical example would be the electric field surrounding a semi-infinite plate. Parabolic coordinates Orthogonal coordinate system Curvilinear coordinates Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. P. 658. ISBN 0-07-043316-X. LCCN 52011515. Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. Pp. 186–187. LCCN 55010911.
Parabolic cylindrical coordinates
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Coordinate surfaces of parabolic cylindrical coordinates. The red parabolic cylinder corresponds to σ=2, whereas the yellow parabolic cylinder corresponds to τ=1. The blue plane corresponds to z =2. These surfaces intersect at the point P (shown as a black sphere), which has Cartesian coordinates roughly (2, -1.5, 2).
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Paraboloidal coordinates
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Paraboloidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional parabolic coordinate system. Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. P. 664. ISBN 0-07-043316-X. LCCN 52011515. Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. Pp. 184–185. LCCN 55010911. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill.
Paraboloidal coordinates
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Coordinate surfaces of the three-dimensional paraboloidal coordinates.
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Oblate spheroidal coordinates
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Thus, the two foci are transformed into a ring of radius a in the x-y plane. Spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length. Spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid of revolution. The azimuthal φ can fall anywhere on a full circle, between ± 180 °. These coordinates are favored below because they are not degenerate; the set of coordinates describes a unique point in Cartesian coordinates. The reverse is also true, in the x-y plane inside the focal ring. The foci of all the ellipses in the x-z plane are located on the x-axis at ±a. Geometrically, the angle ν corresponds to the angle of the asymptotes of the hyperbola. The foci of all the hyperbolae are likewise located on the x-axis at ±a. The coordinates may be calculated from the Cartesian coordinates as follows. Another set of spheroidal coordinates are sometimes used where ζ = sinh μ and ξ = sin ν. The curves of constant ξ are the hyperboloids of revolution. The coordinate ζ is restricted by 0 ≤ ζ < ∞ and ξ is restricted by − 1 ≤ ξ < 1. N will then be an integer. An geometrically intuitive set of oblate spheroidal coordinates are sometimes used, where σ = cosh μ and τ = cos ν.
Oblate spheroidal coordinates
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Prolate spheroidal coordinates
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Rotation about the other axis produces oblate spheroidal coordinates. Spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length. Another example is solving for the electric field generated by two small electrode tips. Other limiting cases include areas generated with a missing segment. The azimuthal ϕ belongs to the interval. The distances from the foci located at = are r ± = x + y 2 + 2 = a. An geometrically intuitive set of prolate spheroidal coordinates are sometimes used, where σ = cosh μ and τ = cos ν. Hence, the curves of constant σ are prolate spheroids, whereas the curves of constant τ are hyperboloids of revolution. The coordinate τ belongs to the interval, whereas the coordinate must be greater than or equal to one. τ have a simple relation to the distances to the foci F 1 and F 2. Thus, the distance to F 1 is a, whereas the distance to F 2 is a. Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. P. 661.
Prolate spheroidal coordinates
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The three coordinate surfaces of prolate spheroidal coordinates. The red prolate spheroid (stretched sphere) corresponds to μ=1, and the blue two-sheet hyperboloid corresponds to ν=45°. The yellow half-plane corresponds to φ=-60°, which is measured relative to the x -axis (highlighted in green). The black sphere represents the intersection point of the three surfaces, which has Cartesian coordinates of roughly (0.831, -1.439, 2.182).
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Elliptic cylindrical coordinates
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Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. These definitions correspond to ellipses and hyperbolae. An alternative and geometrically intuitive set of elliptic coordinates are sometimes used, where σ = cosh μ and τ = cos ν. Hence, the curves of constant σ are ellipses, whereas the curves of constant τ are hyperbolae. The coordinate τ must belong to the interval, whereas the σ coordinate must be greater than or equal to one. The coordinates have a simple relation to the distances to the foci F 1 and F 2. Thus, the distance to F 1 is a, whereas the distance to F 2 is a. A typical example would be the electric field surrounding a flat plate of 2 a. The three-dimensional wave equation, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations. The geometric properties of elliptic coordinates can also be useful. Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. P. 657.
Elliptic cylindrical coordinates
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Coordinate surfaces of elliptic cylindrical coordinates. The yellow sheet is the prism of a half-hyperbola corresponding to ν=-45°, whereas the red tube is an elliptical prism corresponding to μ=1. The blue sheet corresponds to z =1. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (2.182, -1.661, 1.0). The foci of the ellipse and hyperbola lie at x = ±2.0.
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Toroidal coordinates
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Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. The focal ring is also known as the circle. The coordinate ranges are − π < σ ≤ π and τ ≥ 0 ≤ ϕ < 2 π. The centers of the constant - spheres lie along the z - axis, whereas the constant - τ tori are centered in the x y plane. The coordinates may be calculated from the Cartesian coordinates as follows. The 3-variable Laplace equation ∇ 2 Φ = 0 admits solution via separation of variables in toroidal coordinates. Making the substitution Φ = U τ − cos σ A separable equation is then obtained. These Legendre functions are often referred to as toroidal harmonics. Toroidal harmonics have interesting properties. The rest of the toroidal harmonics can be obtained, for instance, by using recurrence relations for associated Legendre functions. Typical examples would be the electric potential and electric field of a conducting torus, or in the degenerate case, an electric current-ring. Again, a separable equation is obtained. For identities relating the toroidal harmonics with argument hyperbolic cosine with those of argument hyperbolic cotangent, see the Whipple formulae. Byerly, W E. An elementary treatise on Fourier's series and spherical, cylindrical, ellipsoidal harmonics, with applications to problems in mathematical physics Ginn & co. pp. 264–266 Arfken G.
Toroidal coordinates
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Bispherical coordinates
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Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, F 2 in bipolar coordinates remain points in the bispherical coordinate system. The surfaces of constant τ are non-intersecting spheres of different radii + 2 = a 2 sinh 2 τ that surround the foci. The centers of the constant - spheres lie along the z - axis, whereas the constant - σ tori are centered in the x y plane. The classic applications of bispherical coordinates are in solving e.g. Laplace's equation, for which bispherical coordinates allow a separation of variables. However, the Helmholtz equation is not separable in bispherical coordinates. A typical example would be the electric field surrounding two conducting spheres of different radii. Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. Pp. 665–666. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. P. 182.
Bispherical coordinates
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Bipolar cylindrical coordinates
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Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular z -direction. The term "bipolar" is often used to describe other curves having two singular points, such as ellipses, Cassini ovals. However, the term bipolar coordinates is never used to describe coordinates associated with those curves, e.g. elliptic coordinates. All these cylinders are parallel to the z - axis. In the z = 0 plane, the centers of the constant - constant - τ cylinders lie on the y and x axes, respectively. A typical example would be the electric field surrounding two parallel cylindrical conductors. Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. Pp. 187–190. LCCN 55010911. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. P. 182.
Bipolar cylindrical coordinates
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Coordinate surfaces of the bipolar cylindrical coordinates. The yellow crescent corresponds to σ, whereas the red tube corresponds to τ and the blue plane corresponds to z =1. The three surfaces intersect at the point P (shown as a black sphere).
105.
Conical coordinates
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Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres and by two families of perpendicular cones, aligned along the z- and x-axes, respectively. In this coordinate system, the Helmholtz equation are separable. The factor for the radius r is one, as in spherical coordinates. The corresponding inverse relations are r = ξ 2 + ψ 2 ϕ = 1 sin ζ arctan θ = ζ. Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. P. 659. ISBN 0-07-043316-X. LCCN 52011515. Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. Pp. 183–184. LCCN 55010911.
Conical coordinates