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 particular coordinate qk is the curve, surface, 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 Laplaces equation or the Helmholtz equation, Laplaces equation is separable in 13 orthogonal coordinate systems, and the Helmholtz equation is separable in 11 orthogonal coordinate systems. Orthogonal coordinates never have off-diagonal terms in their metric tensor and 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 mapping of a standard two-dimensional grid of Cartesian coordinates. A complex number z = x + iy can be formed from the coordinates x and y. However, there are other orthogonal coordinate systems in three dimensions that cannot be obtained by projecting or rotating a system, such as the ellipsoidal coordinates. More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces, in Cartesian coordinates, the basis vectors are fixed. What distinguishes orthogonal coordinates is that, though the basis vectors vary, note that the vectors are not necessarily of equal length. The useful functions known as factors of the coordinates are simply the lengths h i of the basis vectors e ^ i. The scale factors are sometimes called Lamé coefficients, but this terminology is best avoided since some more well known coefficients in linear elasticity carry the same name. Components in the basis are most common in applications for clarity of the quantities. The basis vectors shown above are covariant basis vectors, while a vector is an objective quantity, meaning its identity is independent of any coordinate system, the components of a vector depend on what basis the vector is represented in. Note that the summation symbols Σ and the range, indicating summation over all basis vectors, are often omitted. Vector addition and negation are done component-wise just as in Cartesian coordinates with no complication, extra considerations may be necessary for other vector operations. Note however, that all of these operations assume that two vectors in a field are bound to the same point
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 if the geometry of a problem fits well into a skewed system. For example, solving Laplaces equation in a parallelogram will be easiest when done in appropriately skewed coordinates. The simplest 3D case of a coordinate system is a Cartesian one where one of the axes has been bent by some angle ϕ. For this example, the x axis of a Cartesian coordinate has been bent toward the z axis by ϕ, let e 1, e 2, and e 3 respectively be unit vectors along the x, y, and z axes. Well favor writing quantities with respect to the covariant basis, since the basis vectors are all constant, vector addition and subtraction will simply be familiar component-wise adding and subtraction. Now, let a = ∑ i a i e i and b = ∑ i b i e i where the sums indicate summation over all values of the index. Finally, the curl of a vector is ∇ × a = ∑ i, j, k e k ϵ i j k ∂ a i ∂ q i =1 cos
3.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and 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. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and 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 length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
4.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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 as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
5.
Coordinate system
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a 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. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two 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 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 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, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
6.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient 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, classical 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 used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
7.
Coordinate line
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a 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. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two 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 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 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, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
8.
Cartesian coordinate
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and 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. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and 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 length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
9.
Coordinate surfaces
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a 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. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two 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 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 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, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
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 and this is the intersection between the reference plane and the axis. The third coordinate may be called the height or altitude, longitudinal position and they are sometimes called cylindrical polar coordinates and polar cylindrical coordinates, and 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 azimuth φ is the angle between the direction on the chosen plane and the line from the origin to the projection of P on the plane. The height z is the distance from the chosen plane to the point P. As in polar coordinates, the point with cylindrical coordinates has infinitely many equivalent coordinates, namely and. Moreover, if the radius ρ is zero, the azimuth is arbitrary, in situations where someone wants a unique set of coordinates for each point, one may restrict the radius to be non-negative and the azimuth φ to lie in a specific interval spanning 360°, such as. The notation for cylindrical coordinates is not uniform, the ISO standard 31-11 recommends, where ρ is the radial coordinate, φ the azimuth, and z the height. However, the radius is often denoted r or s, the azimuth by θ or t. In concrete situations, and in many illustrations, a positive angular coordinate is measured counterclockwise as seen from any point with positive height. The cylindrical coordinate system is one of many coordinate systems. The following formulae may be used to convert between them, the arcsin function is the inverse of the sine function, and is assumed to return an angle in the range =. These formulas yield an azimuth φ in the range, for other formulas, see the polar coordinate article. Many modern programming languages provide a function that will compute the correct azimuth φ, in the range, given x and y, for example, this function is called by atan2 in the C programming language, and atan in Common Lisp. In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements, the line element is d r = d ρ ρ ^ + ρ d φ φ ^ + d z z ^. The volume element is d V = ρ d ρ d φ d z, the surface element in a surface of constant radius ρ is d S ρ = ρ d φ d z. The surface element in a surface of constant azimuth φ is d S φ = d ρ d z, the surface element in a surface of constant height z is d S z = ρ d ρ d φ
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 called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal 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, 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 spaces and is then referred to as a hyperspherical coordinate system. To define a coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. These choices determine a plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point P are then defined as follows, the inclination is the angle between the zenith direction and the line segment OP. The azimuth is the angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane. The sign of the azimuth is determined by choosing what is a sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate systems definition, the elevation angle is 90 degrees minus the inclination angle. If the inclination is zero or 180 degrees, the azimuth is arbitrary, if the radius is zero, both azimuth and inclination are arbitrary. In linear algebra, the vector from the origin O to the point P is often called the vector of P. Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of to denote radial distance, inclination, and azimuth, respectively, is common practice in physics, and is specified by ISO standard 80000-2,2009, and earlier in ISO 31-11
12.
Coordinate plane
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a 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. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two 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 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 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, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
13.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of 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 area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all 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. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
14.
Scalar (mathematics)
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A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector, more generally, a vector 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 scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, a vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part, the term is also sometimes used informally to mean a vector, matrix, tensor, or other usually compound value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, the term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix. The word scalar derives from the Latin word scalaris, a form of scala. The English word scale also comes from scala, according to a citation in the Oxford English Dictionary the first recorded usage of the term scalar in English came with W. R. A vector space is defined as a set of vectors, a set of scalars, and a multiplication operation that takes a scalar k. For example, in a space, the scalar multiplication k yields. In a function space, kƒ is the function x ↦ k, the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields. According to a theorem of linear algebra, every vector space has a basis. It follows that every vector space over a scalar field K is isomorphic to a vector space where the coordinates are elements of K. For example, every vector space of dimension n is isomorphic to n-dimensional real space Rn. Alternatively, a vector space V can be equipped with a function that assigns to every vector v in V a scalar ||v||. By definition. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k, a vector space equipped with a norm is called a normed vector space. The norm is defined to be an element of Vs scalar field K. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four operations, thus the rational numbers Q are excluded
15.
Vector (geometric)
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In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the 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, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
16.
Tensor
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In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such include the dot product, the cross product. Geometric vectors, often used in physics and engineering applications, given a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The order of a tensor is the dimensionality of the array needed to represent it, or equivalently, for example, a linear map is represented by a matrix in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a 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 two numbers. The concept enabled an alternative formulation of the 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, an operator is represented in a basis as a two-dimensional square n × n array. The numbers in the array are known as the scalar components of the tensor or simply its components. They are denoted by giving their position in the array, as subscripts and superscripts. For example, the components of an order 2 tensor T could be denoted Tij , whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. The total number of required to identify each component uniquely is equal to the dimension of the array. However, the term generally has another meaning in the context of matrices. Just as the components of a change when we change the basis of the vector space. Each tensor comes equipped with a law that details how the components of the tensor respond to a change of basis
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Tensor analysis
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In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such include the dot product, the cross product. Geometric vectors, often used in physics and engineering applications, given a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The order of a tensor is the dimensionality of the array needed to represent it, or equivalently, for example, a linear map is represented by a matrix in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a 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 two numbers. The concept enabled an alternative formulation of the 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, an operator is represented in a basis as a two-dimensional square n × n array. The numbers in the array are known as the scalar components of the tensor or simply its components. They are denoted by giving their position in the array, as subscripts and superscripts. For example, the components of an order 2 tensor T could be denoted Tij , whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. The total number of required to identify each component uniquely is equal to the dimension of the array. However, the term generally has another meaning in the context of matrices. Just as the components of a change when we change the basis of the vector space. Each tensor comes equipped with a law that details how the components of the tensor respond to a change of basis
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Gradient
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In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a variable, for functions of several variables. The gradient is a function, as opposed to a derivative. If f is a differentiable, real-valued function of several variables, like the derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase 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 vector-valued functions of several variables, a further generalization for a function between Banach spaces is the Fréchet derivative. Consider a room in which the temperature is given by a field, 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 sea level at point is H, the gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%, if, instead, the road goes around the hill at an angle, then it will have a shallower slope. This observation can be stated as follows. If the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when H is differentiable, the dot product of the gradient of H with a unit vector is equal to the directional derivative of H in the direction of that unit vector. The gradient of a function f is denoted ∇f or ∇→f where ∇ denotes the vector differential operator. The notation grad f is commonly used for the gradient. The gradient of f is defined as the vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is
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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. The direction of the curl is the axis of rotation, as determined by the rule. If the vector represents the flow velocity of a moving fluid. A vector field whose curl is zero is called irrotational, the curl is a form of differentiation for vector fields. The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used for curl F and this is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. The name curl was first suggested by James Clerk Maxwell in 1871, the curl of a vector field F, denoted by curl F, or ∇ × F, or rot F, at a point is defined in terms of its projection onto various lines through the point. As such, the curl operator maps continuously differentiable functions f, ℝ3 → ℝ3 to continuous functions g, in fact, it maps Ck functions in ℝ3 to Ck −1 functions in ℝ3. Implicitly, curl is defined by, ⋅ n ^ = d e f lim A →0 where ∮C F · dr is a line integral along the boundary of the area in question, and | A | is the magnitude of the area. Note that the equation for each component, k can be obtained by exchanging each occurrence of a subscript 1,2,3 in cyclic permutation, 1→2, 2→3, and 3→1. If are the Cartesian coordinates and are the coordinates, then h i =2 +2 +2 is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices,3,1,2 →1,2,3 →2,3,1. Suppose the vector field describes the velocity field of a fluid flow, if the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The notation ∇ × F has its origins in the similarities to the 3-dimensional cross product, such notation involving operators is common in physics and algebra. However, in coordinate systems, such as polar-toroidal coordinates. This expands as follows, i + j + k Although expressed in terms of coordinates, equivalently, = e k ε k l m ∇ l F m where ek are the coordinate vector fields. Equivalently, using the derivative, the curl can be expressed as, ∇ × F = ♯ Here ♭ and ♯ are the musical isomorphisms
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Boundary conditions
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In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them, problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems, the analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed and this means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of differential equations is devoted to proving that boundary value problems arising from scientific. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions, boundary value problems are similar to initial value problems. Finding the temperature at all points of a bar with one end kept at absolute zero. If the problem is dependent on both space and time, one could specify the value of the problem at a point for all time or at a given time for all space. Concretely, an example of a value is the problem y ″ + y =0 to be solved for the unknown function y with the boundary conditions y =0, y =2. Without the boundary conditions, the solution to this equation is y = A sin + B cos . From the boundary condition y =0 one obtains 0 = A ⋅0 + B ⋅1 which implies that B =0, from the boundary condition y =2 one finds 2 = A ⋅1 and so A =2. One sees that imposing boundary conditions allowed one to determine a unique solution, a boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of a rod is held at absolute zero. A boundary condition which specifies the value of the derivative of the function is a Neumann boundary condition. For example, if there is a heater at one end of a rod, then energy would be added at a constant rate. If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition. Summary of boundary conditions for the function, y, constants c 0 and c 1 specified by the boundary conditions
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Earth sciences
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Earth science or geoscience is a widely embraced term for the fields of science related to the planet Earth. Earth science can be considered to be a branch of planetary science, there are both reductionist and holistic approaches to Earth sciences. The Earth sciences can include the study of geology, the lithosphere, and the structure of the Earths interior, as well as the atmosphere, hydrosphere. Typically, Earth scientists use tools from geography, chronology, physics, chemistry, biology, Geology describes the rocky parts of the Earths crust and its historic development. Major subdisciplines are mineralogy and petrology, geochemistry, geomorphology, paleontology, stratigraphy, structural geology, engineering geology, geophysics and geodesy investigate the shape of the Earth, its reaction to forces and its magnetic and gravity fields. Geophysicists explore the Earths core and mantle as 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 mineral and petroleum exploration. Soil science covers the outermost layer of the Earths crust that is subject to soil formation processes, major subdisciplines include edaphology and pedology. Ecology covers the interactions between the biota, with their natural environment and this field of study differentiates the study of the Earth, from the study of other planets in the Solar System, the Earth being the only planet teeming with life. Hydrology is a study revolved around the movement, distribution, and quality of the water and involves all the components of the cycle on the earth. Sub-disciplines of hydrology include hydrometeorology, surface hydrology, hydrogeology, watershed science, forest hydrology. Glaciology covers the icy parts of the Earth, atmospheric sciences cover the gaseous parts of the Earth between the surface and the exosphere. Major subdisciplines include meteorology, climatology, atmospheric chemistry, and atmospheric physics, plate tectonics, mountain ranges, volcanoes, and earthquakes are geological phenomena that can be explained in terms of physical and chemical processes in the Earths crust. Beneath the Earths crust lies the mantle which is heated by the decay of heavy elements. The mantle is not quite solid and consists of magma which is in a state of semi-perpetual convection and this convection process causes the lithospheric plates to move, albeit slowly. The resulting process is known as plate tectonics, plate tectonics might be thought of as the process by which the Earth is resurfaced. As the result of spreading, new crust and lithosphere is created by the flow of magma from the mantle to the near surface, through fissures. Through subduction, oceanic crust and lithosphere returns to the convecting mantle, volcanoes result primarily from the melting of subducted crust material. Crust material that is forced into the asthenosphere melts, and some portion of the material becomes light enough to rise to the surface—giving birth to volcanoes
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Cartography
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Cartography is the study and practice of making maps. Combining science, aesthetics, and 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 the maps agenda and select traits of the object to be mapped. This is the concern of map editing, traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. Represent the terrain of the object on flat media. This is the concern of map projections, eliminate characteristics of the mapped object that are not relevant to the maps 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 map design, modern cartography constitutes many theoretical and practical foundations of geographic information systems. The earliest known map is a matter of debate, both because the term map isnt well-defined and because some artifacts that might be maps might actually be something else. A wall painting that might depict the ancient Anatolian city of Çatalhöyük has been dated to the late 7th millennium BCE, the oldest surviving world maps are from 9th century BCE Babylonia. One shows Babylon on the Euphrates, surrounded by Assyria, Urartu and several cities, all, in turn, another depicts Babylon as being north of the world center. The ancient Greeks and Romans created maps since Anaximander in the 6th century BCE, in the 2nd century AD, Ptolemy wrote his treatise on cartography, Geographia. This contained Ptolemys world map – the world known to Western society. As early as the 8th century, Arab scholars were translating the works of the Greek geographers into Arabic, in ancient China, geographical literature dates to the 5th century BCE. The oldest extant Chinese maps come from the State of Qin, dated back to the 4th century BCE, in the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song, a star map on the equidistant cylindrical projection. Early forms of cartography of India included depictions of the pole star and these charts may have been used for navigation. Mappa mundi are the Medieval European maps of the world, approximately 1,100 mappae mundi are known to have survived from the Middle Ages. Of these, some 900 are found illustrating manuscripts and the remainder exist as stand-alone documents, the Arab geographer Muhammad al-Idrisi produced his medieval atlas Tabula Rogeriana in 1154
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Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and 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, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and 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-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
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Quantum mechanics
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Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, 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, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. This phase is known as old quantum theory, according to Planck, each energy element is proportional to its frequency, E = h ν, where h is Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle, with a discrete quantum of energy that was dependent on its frequency. The Copenhagen interpretation of Niels Bohr became widely accepted, in the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons. From Einsteins simple postulation was born a flurry of debating, theorizing, thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927
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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 particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its relation to other forces of nature and it applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century and it introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, with relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves. Max Planck, Hermann Minkowski and others did subsequent work, Einstein developed general relativity between 1907 and 1915, with contributions by many others after 1915. The final form of general relativity was published in 1916, the term theory of relativity was based on the expression relative theory used in 1906 by Planck, who emphasized how the theory uses the principle of relativity. In the discussion section of the 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 significant and necessary tool for theorists and experimentalists in the new fields of physics, nuclear physics. By comparison, general relativity did not appear to be as useful and it seemed to offer little potential for experimental test, as most of its assertions were on an astronomical scale. Its mathematics of general relativity seemed difficult and fully understandable only by a number of people. Around 1960, general relativity became central to physics and astronomy, new mathematical techniques to apply to general relativity streamlined calculations and made its concepts more easily visualized. Special relativity is a theory of the structure of spacetime and it was introduced in Einsteins 1905 paper On the Electrodynamics of Moving Bodies. Special relativity is based on two postulates which are contradictory in classical mechanics, The laws of physics are the same for all observers in motion relative to one another. The speed of light in a vacuum is the same for all observers, the resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment, moreover, the theory has many surprising and counterintuitive consequences. Some of these are, Relativity of simultaneity, Two events, simultaneous for one observer, time dilation, Moving clocks are measured to tick more slowly than an observers stationary clock
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Engineering
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The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, 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, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
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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 meaning of the term dimension. In physics and 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 three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
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Position vector
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Usually denoted x, r, or s, it corresponds to the straight-line distances along each axis from O to P, r = O P →. The term position vector is used mostly in the fields of geometry, mechanics. Frequently this is used in two-dimensional or three-dimensional space, but can be generalized to Euclidean spaces in any number of dimensions. These different coordinates and corresponding basis vectors represent the position vector. More general curvilinear coordinates could be used instead, and are in contexts like continuum mechanics, linear algebra allows for the abstraction of an n-dimensional position vector. The notion of space is intuitive since each xi can be any value, the dimension of the position space is n. The coordinates of the vector r with respect to the vectors ei are xi. The vector of coordinates forms the coordinate vector or n-tuple, each coordinate xi may be parameterized a number of parameters t. One parameter xi would describe a curved 1D path, two parameters xi describes a curved 2D surface, three xi describes a curved 3D volume of space, and so on. The linear span of a basis set B = equals the position space R, position vector fields are used to describe continuous and differentiable space curves, in which case the independent parameter needs not be time, but can be arc length of the curve. In the case of one dimension, the position has only one component and it could be, say, a vector in the x-direction, or the radial r-direction. Equivalent notations include, x ≡ x ≡ x, r ≡ r, s ≡ s ⋯ For a position vector r that is a function of time t and these derivatives have common utility in the study of kinematics, control theory, engineering and other sciences. Velocity v = d r d t where dr is a small displacement. By extension, the higher order derivatives can be computed in a similar fashion, study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite sequence, enabling several analytical techniques in engineering. A displacement vector can be defined as the action of uniformly translating spatial points in a given direction over a given distance, thus the addition of displacement vectors expresses the composition of these displacement actions and scalar multiplication as scaling of the distance. With this in mind we may define a position vector of a point in space as the displacement vector mapping a given origin to that point. Note thus position vectors depend on a choice of origin for the space, affine space Six degrees of freedom Line element Parametric surface Keller, F. J, Gettys, W. E. et al
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Standard basis
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In mathematics, the standard basis for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system. For example, the basis for the Euclidean plane is formed by vectors e x =, e y =. Here the vector ex points in the x direction, the vector ey points in the y direction, there are several common notations for these vectors, including, and. These vectors are 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 and these vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z, the scalars vx, vy, vz being the scalar components of the vector v. In n -dimensional Euclidean space, the standard consists of n distinct vectors. Standard bases can be defined for vector spaces, such as polynomials. In both cases, the standard consists of the elements of the vector space such that all coefficients but one are 0. For polynomials, the standard basis consists of the monomials and is commonly called monomial basis. For matrices M m × n, the standard consists of the m×n-matrices with exactly one non-zero entry. For example, the basis for 2×2 matrices is formed by the 4 matrices e 11 =, e 12 =, e 21 =, e 22 =. By definition, the 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. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i. e, there is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials. This family is the basis of the R-module R of all families f = from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R. The existence of standard bases has become a topic of interest in algebraic geometry. It is now a part of theory called standard monomial theory
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Basis (linear algebra)
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In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space V, every element of V can be expressed uniquely as a combination of basis vectors. A vector space can have distinct sets of basis vectors, however each such set has the same number of elements. A basis B of a vector space V over a field F is an independent subset of V that spans V. In more detail, suppose that B = is a subset of a vector space V over a field F. The numbers ai are called the coordinates of the vector x with respect to the basis B, a vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the definition to include infinite basis sets. The sums in the definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept and it is often convenient to list the basis vectors in a specific order, for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of a basis, which we define to be a sequence of linearly independent vectors that span V. B is a set of linearly independent vectors, i. e. it is a linearly independent set. Every vector in V can be expressed as a 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 relative to the basis. Every vector space has a basis, the proof of this requires the axiom of choice. All bases of a vector space have the same cardinality, called the dimension of the vector space and this result is known as the dimension theorem, and requires the ultrafilter lemma, 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 ith column of the identity matrix. In P2, where P2 is the set of all polynomials of degree at most 2, is the standard basis. In M22, where M22 is the set of all 2×2 matrices. and Mm, n is the 2×2 matrix with a 1 in the m, n position, given a vector space V over a field F and suppose that and are two bases for V
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 to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a 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 geometry and has been extensively generalized. 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 other straight line could fall between it and 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 and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
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 and 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, so, an element of T M can be thought of\as a pair, where x is a point in M and v is a tangent vector to M at x. There is a natural projection π, T M ↠ M defined by π = x and this projection maps each tangent space T x M to the single point x. The tangent bundle comes equipped with a natural topology, with this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle. A section of T M is a field on M, and the dual bundle to T M is the cotangent bundle. By definition, a manifold M is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, for example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n=1,3,7. One of the roles of the tangent bundle is to provide a domain. Namely, if f, M → N is a function, with M and N smooth manifolds, its derivative is a smooth function Df. The tangent bundle comes equipped with a 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. If U is an open subset of M, then there is a diffeomorphism from TU to U × Rn which restricts to a linear isomorphism from each tangent space TxU to × Rn. As a manifold, however, TM is not always diffeomorphic to the product manifold M × Rn, when it is of the form M × Rn, then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur for manifolds equipped with a group structure, for instance. The tangent bundle of the circle is trivial because it is a Lie group. It is not true however that all spaces with trivial tangent bundles are Lie groups, just as manifolds are locally modelled on Euclidean space, tangent bundles are locally modelled on U × Rn, where U is an open subset of Euclidean space. If M is a smooth manifold, then it comes equipped with an atlas of charts where Uα is an open set in M and ϕ α, U α → R n is a diffeomorphism
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 range of applications, including for mechanical engineering, civil engineering, chemical engineering, geophysics, astrophysics. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, fluid mechanics, especially fluid dynamics, is an active field of research with many problems that are partly or wholly unsolved. Fluid mechanics can be complex, and can best be solved by numerical methods. A modern discipline, called computational fluid dynamics, is devoted to this approach to solving fluid mechanics problems, Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow. Inviscid flow was further analyzed by mathematicians and viscous flow was explored by a multitude of engineers including Jean Léonard Marie Poiseuille. Fluid statics or hydrostatics is the branch of mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium, and is contrasted with fluid dynamics, hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to some aspect of geophysics and astrophysics, to meteorology, to medicine, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the science of liquids and gases in motion. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density and it has several subdisciplines itself, including aerodynamics and hydrodynamics. Some fluid-dynamical principles are used in engineering and crowd dynamics. Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table, in a mechanical view, a fluid is a substance that does not support shear stress, that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress, the assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. This can be expressed as an equation in integral form over the control volume, the continuum assumption is an idealization of continuum mechanics under which fluids can be treated as continuous, even though, on a microscopic scale, they are composed of molecules. Fluid properties can vary continuously from one element to another and are average values of the molecular properties. The continuum hypothesis can lead to results in applications like supersonic speed flows. Those problems for which the continuum hypothesis fails, can be solved using statistical mechanics, to determine whether or not the continuum hypothesis applies, the Knudsen number, defined as the ratio of the molecular mean free path to the characteristic length scale, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, the Navier–Stokes equations are differential equations that describe the force balance at a given point within a fluid
34.
Continuum mechanics
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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century, research in the area continues till 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 solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are 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, liquids and gases, are composed of molecules separated by space. On a microscopic scale, materials have cracks and discontinuities, a continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material. More specifically, the continuum hypothesis/assumption hinges on the concepts of an elementary volume. This condition provides a link between an experimentalists and a viewpoint on constitutive equations as well as a way of spatial and statistical averaging of the microstructure. The latter then provide a basis for stochastic 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 car traffic on a highway---with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a differential equation for the density of cars. The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. To start modelling define that, x measure distance along the highway, t is time, ρ is the density of cars on the highway, cars do not appear and disappear. Consider any group of cars, from the car at the back of the group located at x = a to the particular car at the front located at x = b. The total number of cars in this group N = ∫ a b ρ d x, since cars are conserved d N / d t =0. The only way an integral can be zero for all intervals is if the integrand is zero for all x, consequently, conservation derives the first order nonlinear conservation PDE ∂ ρ ∂ t + ∂ ∂ x =0 for all positions on the highway. This conservation PDE applies not only to car traffic but also to fluids, solids, crowds, animals, plants, bushfires, financial traders and this PDE is one equation with two unknowns, so another equation is needed to form a well posed problem
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. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century, 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 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, Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. These unanswered questions indicated greater, hidden relationships, initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric and this is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Various concepts based on length, such as the arc length of curves, area of plane regions, the notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. 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. These are the closest analogues to the plane and space considered in Euclidean and non-Euclidean geometry. Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite, a special case of this is a Lorentzian manifold, which is the mathematical basis of Einsteins general relativity theory of gravity. Finsler geometry has the Finsler manifold as the object of study. This is a manifold with a Finsler metric, i. e. a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M is a function F, TM → [0, ∞) such that, F = |m|F for all x, y in TM, F is infinitely differentiable in TM −, symplectic geometry is the study of symplectic manifolds. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed, a diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, in dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism
36.
Total differential
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In calculus, the differential represents the principal part of the change in a function y = f with respect to changes in the independent variable. The differential dy is defined by d y = f ′ d x, where f ′ is the derivative of f with respect to x, one also writes d f = f ′ d x. The precise meaning of the variables dy and dx depends on the context of the application, traditionally, the variables dx and dy are considered to be very small, and this interpretation is made rigorous in non-standard analysis. The quotient 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 famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential without appeal to the atomism of Leibnizs infinitesimals, in physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. Courant & John reconcile the use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the purpose for which they are intended. Thus physical infinitesimals need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense, following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real analysis, it is 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 functional of an increment Δx. This approach allows the differential to be developed for a variety of more sophisticated spaces, 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 calculus as follows. The differential of a function f of a real variable x is the function df of two independent real variables x and Δx given by d f = d e f f ′ Δ x. One or both of the arguments may be suppressed, i. e. one may see df or simply df, if y = f, the differential may also be written as dy. The partial differential is therefore ∂ y ∂ x 1 d x 1 involving the partial derivative of y with respect to x1. The total differential is then defined as d y = ∂ y ∂ x 1 Δ x 1 + ⋯ + ∂ y ∂ x n Δ x n. Since, with this definition, d x i = Δ x i, in measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy. of the parameters x, y. As they are assumed to be independent, the analysis describes the worst-case scenario, the absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign
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. For instance, in changing scale from meters to centimeters, the components of a velocity vector will multiply by 100. Vectors exhibit this behavior of changing scale inversely to changes in scale to the reference axes, 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 units of a spatial derivative, or distance−1. The components of dual vectors change in the way as changes to scale of the reference axes. 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 matrix as the change of basis matrix. The components of vectors are said to be covariant. Examples of covariant vectors generally appear when taking a gradient of a function, in Einstein notation, covariant components are denoted with lower indices as in v = v i e i. Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are used in physical. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance, in physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list of numbers such as. The numbers in the list depend on the choice of coordinate system, for a vector to represent a geometric object, it must be possible to describe how it looks in any other coordinate system. That is to say, the components of the vectors will transform in a way in passing from one coordinate system to another. A contravariant vector has components that transform as the coordinates do under changes of coordinates, including rotation and dilation. The vector itself does not change under these operations, instead, in other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction and this important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities
38.
Del
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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 domain, it denotes its standard derivative as defined in calculus. When applied to a field, del may denote the gradient of a scalar field, strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators, that makes many equations easier to write and remember. These formal products do not necessarily commute with other operators or products, del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian. In particular, if a hill is defined as a function over a plane h. The magnitude of the gradient is the value of this steepest slope, when operating on a vector it must be distributed to each component. The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplaces equation, Poissons equation, the equation, the wave equation. Del can also be applied to a field with the result being a tensor. The tensor derivative of a vector field v → is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ∇ ⊗ v →, where ⊗ represents the dyadic product. 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, divergence, and curl and this is part of the value to be gained in notationally representing this operator as a vector. Though one can often replace del with a vector and obtain an identity, making those identities mnemonic. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function. For that reason, identities involving del must be derived with care, schey, H. M. Div, Grad, Curl, and All That, An Informal Text on Vector Calculus. Earliest Uses of Symbols of Calculus, NA Digest, Volume 98, Issue 03. A survey of the use of ∇ in vector analysis Tai
39.
Linear operator
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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 case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category 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. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE