1.
Euclidean vector
–
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
2.
Space (mathematics)
–
In mathematics, a space is a set with some added structure. Mathematical spaces often form a hierarchy, i. e. one space may inherit all the characteristics of a parent space, modern mathematics treats space quite differently compared to classical mathematics. In the ancient mathematics, space was an abstraction of the three-dimensional space observed in the everyday life. The axiomatic method had been the research tool since Euclid. The method of coordinates was adopted by René Descartes in 1637, two equivalence relations between geometric figures were used, congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures, homotheties — into similar figures, for example, all circles are mutually similar, but ellipses are not similar to circles. The relation between the two geometries, Euclidean and projective, shows that objects are not given to us with their structure. Rather, each mathematical theory describes its objects by some of their properties, distances and angles are never mentioned in the axioms of the projective geometry and therefore cannot appear in its theorems. The question what is the sum of the three angles of a triangle is meaningful in the Euclidean geometry but meaningless in the projective geometry. A different situation appeared in the 19th century, in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value. The non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829, eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean models of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory. This discovery forced the abandonment of the pretensions to the truth of Euclidean geometry. It showed that axioms are not obvious, nor implications of definitions, to what extent do they correspond to an experimental reality. This important physical problem no longer has anything to do with mathematics, even if a geometry does not correspond to an experimental reality, its theorems remain no less mathematical truths. These Euclidean objects and relations play the non-Euclidean geometry like contemporary actors playing an ancient performance, relations between the actors only mimic relations between the characters in the play. Likewise, the relations between the chosen objects of the Euclidean model only mimic the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not, according to Nicolas Bourbaki, the period between 1795 and 1872 can be called the golden age of geometry. Analytic geometry made a progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups
3.
Force
–
In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
4.
Magnetic field
–
A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any point is specified by both a direction and a magnitude, as such it is represented by a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter in the SI, B is measured in teslas and newtons per meter per ampere in the SI. B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges, Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In quantum physics, the field is quantized and electromagnetic interactions result from the exchange of photons. Magnetic fields are used throughout modern technology, particularly in electrical engineering. The Earth produces its own field, which is important in navigation. Rotating magnetic fields are used in electric motors and generators. Magnetic forces give information about the carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits, noting that the resulting field lines crossed at two points he named those points poles in analogy to Earths poles. He also clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them, almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilberts work, De Magnete, helped to establish magnetism as a science, in 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated, building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by magnetic poles, three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Ørsted discovered that an electric current generates a magnetic field encircling it, then in 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot and Félix Savart discovered the Biot–Savart law in 1820, extending these experiments, Ampère published his own successful model of magnetism in 1825. This has the benefit of explaining why magnetic charge can not be isolated. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism, in 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field
5.
Gravity
–
Gravity, or gravitation, is a natural phenomenon by which all things with mass are brought toward one another, including planets, stars and galaxies. Since energy and mass are equivalent, all forms of energy, including light, on Earth, gravity gives weight to physical objects and causes the ocean tides. Gravity has a range, although its effects become increasingly weaker on farther objects. The most extreme example of this curvature of spacetime is a hole, from which nothing can escape once past its event horizon. More gravity results in time dilation, where time lapses more slowly at a lower gravitational potential. Gravity is the weakest of the four fundamental interactions of nature, the gravitational attraction is approximately 1038 times weaker than the strong force,1036 times weaker than the electromagnetic force and 1029 times weaker than the weak force. As a consequence, gravity has an influence on the behavior of subatomic particles. On the other hand, gravity is the dominant interaction at the macroscopic scale, for this reason, in part, pursuit of a theory of everything, the merging of the general theory of relativity and quantum mechanics into quantum gravity, has become an area of research. While the modern European thinkers are credited with development of gravitational theory, some of the earliest descriptions came from early mathematician-astronomers, such as Aryabhata, who had identified the force of gravity to explain why objects do not fall out when the Earth rotates. Later, the works of Brahmagupta referred to the presence of force, described it as an attractive force. Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and this was a major departure from Aristotles belief that heavier objects have a higher gravitational acceleration. Galileo postulated air resistance as the reason that objects with less mass may fall slower in an atmosphere, galileos work set the stage for the formulation of Newtons theory of gravity. In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. Newtons theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the position of the planet. A discrepancy in Mercurys orbit pointed out flaws in Newtons theory, the issue was resolved in 1915 by Albert Einsteins new theory of general relativity, which accounted for the small discrepancy in Mercurys orbit. The simplest way to test the equivalence principle is to drop two objects of different masses or compositions in a vacuum and see whether they hit the ground at the same time. Such experiments demonstrate that all objects fall at the rate when other forces are negligible
6.
Calculus
–
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
7.
Work (physics)
–
In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the SI unit of work is the joule. The SI unit of work is the joule, which is defined as the work expended by a force of one newton through a distance of one metre. The dimensionally equivalent newton-metre is sometimes used as the unit for work, but this can be confused with the unit newton-metre. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton metres is a torque measurement, or a measurement of energy. Non-SI units of work include the erg, the foot-pound, the foot-poundal, the hour, the litre-atmosphere. Due to work having the physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU. The work done by a constant force of magnitude F on a point that moves a distance s in a line in the direction of the force is the product W = F s. For example, if a force of 10 newtons acts along a point that travels 2 meters and this is approximately the work done lifting a 1 kg weight from ground level to over a persons head against the force of gravity. Notice that the work is doubled either by lifting twice the weight the distance or by lifting the same weight twice the distance. Work is closely related to energy, the work-energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in energy is caused by an equal amount of negative work done by the resultant force. From Newtons second law, it can be shown that work on a free, rigid body, is equal to the change in energy of the velocity and rotation of that body. The work of forces generated by a function is known as potential energy. These formulas demonstrate that work is the associated with the action of a force, so work subsequently possesses the physical dimensions. The work/energy principles discussed here are identical to Electric work/energy principles, constraint forces determine the movement of components in a system, constraining the object within a boundary. Constraint forces ensure the velocity in the direction of the constraint is zero and this only applies for a single particle system. For example, in an Atwood machine, the rope does work on each body, there are, however, cases where this is not true
8.
Conservation of energy
–
In physics, the law of conservation of energy states that the total energy of an isolated system remains constant—it is said to be conserved over time. Energy can neither be created nor destroyed, rather, it transforms from one form to another, for instance, chemical energy can be converted to kinetic energy in the explosion of a stick of dynamite. A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist and that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings. Ancient philosophers as far back as Thales of Miletus c.550 BCE had inklings of the conservation of some underlying substance of everything is made. However, there is no reason to identify this with what we know today as mass-energy. Empedocles wrote that in his system, composed of four roots, nothing comes to be or perishes, instead. In 1605, Simon Stevinus was able to solve a number of problems in statics based on the principle that perpetual motion was impossible. Essentially, he pointed out that the height a moving body rises is equal to the height from which it falls, and used this observation to infer the idea of inertia. The remarkable aspect of this observation is that the height that a moving body ascends to does not depend on the shape of the surface that the body is moving on. In 1669, Christian Huygens published his laws of collision, among the quantities he listed as being invariant before and after the collision of bodies were both the sum of their linear momentums as well as the sum of their kinetic energies. However, the difference between elastic and inelastic collision was not understood at the time and this led to the dispute among later researchers as to which of these conserved quantities was the more fundamental. In his Horologium Oscillatorium, he gave a much clearer statement regarding the height of ascent of a moving body, Huygens study of the dynamics of pendulum motion was based on a single principle, that the center of gravity of heavy objects cannot lift itself. The fact that energy is scalar, unlike linear momentum which is a vector. It was Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with motion. Using Huygens work on collision, Leibniz noticed that in mechanical systems. He called this quantity the vis viva or living force of the system, the principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Many physicists at that time, such as Newton, held that the conservation of momentum and it was later shown that both quantities are conserved simultaneously, given the proper conditions such as an elastic collision. In 1687, Isaac Newton published his Principia, which was organized around the concept of force and momentum
9.
Fundamental theorem of calculus
–
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the functions integral. This part of the guarantees the existence of antiderivatives for continuous functions. This part of the theorem has key practical applications because it simplifies the computation of definite integrals. The fundamental theorem of calculus relates differentiation and integration, showing that two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration, the first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory. Isaac Barrow proved a more generalized version of the theorem, while his student Isaac Newton completed the development of the mathematical theory. Gottfried Leibniz systematized the knowledge into a calculus for infinitesimal quantities, for a continuous function y = f whose graph is plotted as a curve, each value of x has a corresponding area function A, representing the area beneath the curve between 0 and x. The function A may not be known, but it is given that it represents the area under the curve. The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x, in other words, the area of this “sliver” would be A − A. There is another way to estimate the area of this same sliver, as shown in the accompanying figure, h is multiplied by f to find the area of a rectangle that is approximately the same size as this sliver. So, A − A ≈ f h In fact, this becomes a perfect equality if we add the red portion of the excess area shown in the diagram. So, A − A = f h + Rearranging terms, as h approaches 0 in the limit, the last fraction can be shown to go to zero. This is true because the area of the red portion of region is less than or equal to the area of the tiny black-bordered rectangle. More precisely, | f − A − A h | = | Red Excess | h ≤ h | f − f | h = | f − f |, by the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does and that is, the derivative of the area function A exists and is the original function f, so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and “finding the area” under its curve are opposite operations and this is the crux of the Fundamental Theorem of Calculus. Intuitively, the theorem states that the sum of infinitesimal changes in a quantity over time adds up to the net change in the quantity
10.
Curl (mathematics)
–
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
11.
Euclidean space
–
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
12.
Vector-valued function
–
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a function could be a scalar or a vector. The dimension of the domain is not defined by the dimension of the range, a common example of a vector-valued function is one that depends on a single real number parameter t, often representing time, producing a vector v as the result. The vector r has its tail at the origin and its head at the coordinates evaluated by the function, the vector shown in the graph to the right is the evaluation of the function near t=19.5. The spiral is the path traced by the tip of the vector as t increases from zero through 8π, many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system. Thus, if r = f i + g j + h k is a vector-valued function, the vector derivative admits the following physical interpretation, if r represents the position of a particle, then the derivative is the velocity of the particle v = d r d t. Likewise, the derivative of the velocity is the acceleration d v d t = a and it is also called the direction cosine of a and ei or their dot product. The vectors e1, e2, e3 form an orthonormal basis fixed in the frame in which the derivative is being taken. Some authors prefer to use capital D to indicate the total derivative operator, the total derivative differs from the partial time derivative in that the total derivative accounts for changes in a due to the time variance of the variables qr. Whereas for scalar-valued functions there is only a possible reference frame. Once a reference frame has been chosen, the derivative of a function can be computed using techniques similar to those for computing derivatives of scalar-valued functions. A different choice of frame will, in general, produce a different derivative function. The derivative functions in different reference frames have a specific kinematical relationship and this often holds true for problems dealing with vector fields in a fixed coordinate system, or for simple problems in physics. However, many problems involve the derivative of a vector function in multiple moving reference frames. As shown previously, the first term on the hand side is equal to the derivative of a in the reference frame where e1, e2, e3 are constant. It also can be shown that the term on the right hand side is equal to the relative angular velocity of the two reference frames cross multiplied with the vector a itself. Since velocity is the derivative of position, NvR and EvR are the derivatives of rR in reference frames N and E, respectively. By substitution, N v R = E v R + N ω E × r R where EvR is the velocity vector of the rocket as measured from a reference frame E that is fixed to the Earth
13.
Covariance and contravariance of vectors
–
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
14.
Open set
–
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. These conditions are very loose, and they allow enormous flexibility in the choice of open sets, in the two extremes, every set can be open, or no set can be open but the space itself and the empty set. In practice, however, open sets are usually chosen to be similar to the intervals of the real line. The notion of an open set provides a way to speak of nearness of points in a topological space. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of importance in point-set topology. Intuitively, an open set provides a method to distinguish two points, for example, if about one point in a topological space there exists an open set not containing another point, the two points are referred to as topologically distinguishable. In this manner, one may speak of two subsets of a topological space are near without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces, in the set of all real numbers, one has the natural Euclidean metric, that is, a function which measures the distance between two real numbers, d = |x - y|. Therefore, given a number, one can speak of the set of all points close to that real number. In essence, points within ε of x approximate x to an accuracy of degree ε, note that ε >0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x =0 and ε =1, the points within ε of x are precisely the points of the interval, that is, however, with ε =0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a degree of accuracy compared to when ε =1. The previous discussion shows, for the case x =0, in particular, sets of the form give us a lot of information about points close to x =0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x, thus, we find that in some sense, every real number is distance 0 away from 0. It may help in case to think of the measure as being a binary condition, all things in R are equally close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis, in fact, one may generalize these notions to an arbitrary set, rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets around x, of course, this collection would have to satisfy certain properties for otherwise we may not have a well-defined method to measure distance
15.
Surface (topology)
–
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an abstract surface not embedded in any Euclidean space. For example, the Klein bottle is a surface, which cannot be represented in the three-dimensional Euclidean space without introducing self-intersections, in mathematics, a surface is a geometrical shape that resembles to a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, the exact definition of a surface may depend on the context. Typically, in geometry, a surface may cross itself, while, in topology and differential geometry. A surface is a space, this means that a moving point on a surface may move in two directions. In other words, around almost every point, there is a patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, the concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the properties of an airplane. A surface is a space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a chart and it is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean. In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second countable. It is also assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second countable and these homeomorphisms are also known as charts. The boundary of the upper half-plane is the x-axis, a point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of points is known as the boundary of the surface which is necessarily a one-manifold, that is. On the other hand, a point mapped to above the x-axis is an interior point, the collection of interior points is the interior of the surface which is always non-empty. The closed disk is an example of a surface with boundary
16.
Differential geometry of curves
–
This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian manifolds, for a discussion of curves in an arbitrary topological space, see the main article on curves. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane, starting in antiquity, many concrete curves have been thoroughly investigated using the synthetic approach. Any regular curve may be parametrized by the arc length and from the point of view of a bug on the curve that does not know anything about the ambient space, different space curves are only distinguished by the way in which they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the curvature, the fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve. Let n ∈ N, r ∈ N ∪, and I be a non-empty interval of real numbers, then a vector-valued function γ, I → R n of class C r is called a parametric C r -curve or a C r -parametrization. Note that γ ⊆ R n is called the image of the parametric curve and it is important to distinguish between a parametric curve γ and its image γ, because a given subset of R n can be the image of several distinct parametric curves. One may think of the t in γ as representing time. If I is an interval, then we call γ the starting point. If γ = γ, then we say that γ is closed or is a loop, furthermore, we call γ a closed parametric C r -curve if and only if γ = γ for all k ∈ N ≤ r. If γ |, → R n is injective, then we say that γ is simple, if each component function of γ, I → R can be locally expressed as a power series, then we say that γ is analytic or of class C ω. We write − γ for the curve that is traversed in the direction opposite to that of γ. We say that γ is regular of order m if and only if for any t ∈ I, is an independent subset of R n. In particular, a parametric C1 -curve γ is regular if, given the image of a parametric curve, one can define several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of curves that are invariant under certain re-parametrizations. We thus have to define an equivalence relation on the set of all parametric curves. The differential-geometric properties of a curve are invariant under re-parametrization. The equivalence classes are called C r -curves and are central objects studied in the geometry of curves
17.
Differentiable manifold
–
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas, one may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart, in formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. In other words, where the domains of overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the charts to one another are called transition maps. Differentiability means different things in different contexts including, continuously differentiable, k times differentiable, smooth, furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for theories such as classical mechanics, general relativity. It is possible to develop a calculus for differentiable manifolds and this leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry, the emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen and these ideas found a key application in Einsteins theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces, the widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space. This formalizes the notion of patching together pieces of a space to make a manifold – the manifold produced also contains the data of how it has been patched together, However, different atlases may produce the same manifold, a manifold does not come with a preferred atlas. And, thus, one defines a manifold to be a space as above with an equivalence class of atlases. There are a number of different types of manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following, a differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable
18.
Section (fiber bundle)
–
In the mathematical field of topology, a section of a fiber bundle E is a continuous right inverse of the projection function π. A section is an abstract characterization of what it means to be a graph, then a graph is any function σ for which π = x. The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product, if π, E → B is a fibre bundle, then a section is a choice of point σ in each of the fibres. The condition π = x simply means that the section at a point x must lie over x, for example, when E is a vector bundle a section of E is an element of the vector space Ex lying over each point x ∈ B. Sections, particularly of principal bundles and vector bundles, are very important tools in differential geometry. In this setting, the base space B is a smooth manifold M, in this case, one considers the space of smooth sections of E over an open set U, denoted C∞. It is also useful in analysis to consider spaces of sections with intermediate regularity. Fiber bundles do not in general have such sections, so it is also useful to define sections only locally. A local section of a bundle is a continuous map s, U → E where U is an open set in B and π = x for all x in U. If is a trivialization of E, where φ is a homeomorphism from π−1 to U × F. The sections form a sheaf over B called the sheaf of sections of E, the space of continuous sections of a fiber bundle E over U is sometimes denoted C, while the space of global sections of E is often denoted Γ or Γ. Sections are studied in homotopy theory and algebraic topology, where one of the goals is to account for the existence or non-existence of global sections. An obstruction denies the existence of global sections since the space is too twisted, more precisely, obstructions obstruct the possibility of extending a local section to a global section due to the spaces twistedness. Obstructions are indicated by particular characteristic classes, which are cohomological classes, for example, a principal bundle has a global section if and only if it is trivial. On the other hand, a vector bundle always has a global section, however, it only admits a nowhere vanishing section if its Euler class is zero. Obstructions to extending local sections may be generalized in the manner, take a topological space and form a category whose objects are open subsets. Thus we use a category to generalize a topological space and we generalize the notion of a local section using sheaves of abelian groups, which assigns to each object an abelian group. There is an important distinction here, intuitively, local sections are like vector fields on a subset of a topological space
19.
Tangent bundle
–
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
20.
Differentiable function
–
In calculus, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a function must have a tangent line at each point in its domain, be relatively smooth. More generally, if x0 is a point in the domain of a function f and this means that the graph of f has a non-vertical tangent line at the point. The function f may also be called locally linear at x0, if f is differentiable at a point x0, then f must also be continuous at x0. In particular, any function must be continuous at every point in its domain. The converse does not hold, a function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a set in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions, the first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. A function f is said to be continuously differentiable if the derivative f exists and is itself a continuous function, though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function f = { x 2 sin if x ≠00 if x =0 is differentiable at 0, since f ′ = lim ϵ →0 =0, exists. However, for x≠0, f ′ =2 x sin − cos which has no limit as x →0, nevertheless, Darbouxs theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. Sometimes continuously differentiable functions are said to be of class C1, a function is of class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives f′, F all exist and are continuous. If derivatives f exist for all integers n, the function is smooth or equivalently. If all the derivatives of a function exist and are continuous in a neighborhood of a point, then the function is differentiable at that point. If a function is differentiable at x0, then all of the partial derivatives exist at x0, a similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. Note that existence of the partial derivatives does not in general guarantee that a function is differentiable at a point
21.
Ring (mathematics)
–
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity
22.
Linear form
–
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. The set of all linear functionals from V to k, Homk, forms a space over k with the addition of the operations of addition. This space is called the space of V, or sometimes the algebraic dual space. It is often written V∗ or V′ when the field k is understood, if V is a topological vector space, the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If V is a Banach space, then so is its dual, to distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual. In finite dimensions, every linear functional is continuous, so the dual is the same as the algebraic dual. Suppose that vectors in the coordinate space Rn are represented as column vectors x → =. For each row there is a linear functional f defined by f = a 1 x 1 + ⋯ + a n x n. This is just the product of the row vector and the column vector x →, f =. Linear functionals first appeared in functional analysis, the study of spaces of functions. Let Pn denote the space of real-valued polynomial functions of degree ≤n defined on an interval. If c ∈, then let evc, Pn → R be the evaluation functional, the mapping f → f is linear since = f + g = α f. If x0, …, xn are n+1 distinct points in, then the evaluation functionals evxi, the integration functional I defined above defines a linear functional on the subspace Pn of polynomials of degree ≤ n. If x0, …, xn are n+1 distinct points in, then there are coefficients a0, … and this forms the foundation of the theory of numerical quadrature. This follows from the fact that the linear functionals evxi, f → f defined above form a basis of the space of Pn. Linear functionals are particularly important in quantum mechanics, quantum mechanical systems are represented by Hilbert spaces, which are anti–isomorphic to their own dual spaces. A state of a mechanical system can be identified with a linear functional. For more information see bra–ket notation, in the theory of generalized functions, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functions
23.
Scalar field
–
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a number or a physical quantity. Examples used in include the temperature distribution throughout space, the pressure distribution in a fluid. These fields are the subject of field theory. Mathematically, a field on a region U is a real or complex-valued function or distribution on U. A scalar field is a field of order zero. Physically, a field is additionally distinguished by having units of measurement associated with it. Scalar fields are contrasted with other physical quantities such as vector fields, more subtly, scalar fields are often contrasted with pseudoscalar fields. In physics, scalar fields often describe the energy associated with a particular force. The force is a field, which can be obtained as the gradient of the potential energy scalar field. Examples include, Potential fields, such as the Newtonian gravitational potential, a temperature, humidity or pressure field, such as those used in meteorology. In quantum field theory, a field is associated with spin-0 particles. The scalar field may be real or complex valued, complex scalar fields represent charged particles. These include the charged Higgs field of the Standard Model, as well as the charged pions mediating the nuclear interaction. This mechanism is known as the Higgs mechanism, a candidate for the Higgs boson was first detected at CERN in 2012. In scalar theories of gravitation scalar fields are used to describe the gravitational field, scalar-tensor theories represent the gravitational interaction through both a tensor and a scalar. Such attempts are for example the Jordan theory as a generalization of the Kaluza–Klein theory, scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model. This field interacts gravitationally and Yukawa-like with the particles that get mass through it, scalar fields are found within superstring theories as dilaton fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor
24.
Sphere
–
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
25.
Tangent space
–
The elements of the tangent space are called tangent vectors at x. This is a generalization of the notion of a vector in a Euclidean space. The dimension of all the tangent spaces of a manifold is the same as that of the manifold. More generally, if a manifold is thought of as an embedded submanifold of Euclidean space one can picture a tangent space in this literal fashion. This was the approach to defining parallel transport, and used by Dirac. More strictly this defines a tangent space, distinct from the space of tangent vectors described by modern terminology. In algebraic geometry, in contrast, there is a definition of tangent space at a point P of a variety V. The points P at which the dimension is exactly that of V are called the non-singular points, for example, a curve that crosses itself doesnt have a unique tangent line at that point. The singular points of V are those where the test to be a manifold fails, once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point, all the tangent spaces can be glued together to form a new differentiable manifold of twice the dimension of the original manifold, called the tangent bundle of the manifold. The informal description above relies on a manifold being embedded in a vector space Rm. However, it is convenient to define the notion of tangent space based on the manifold itself. There are various equivalent ways of defining the tangent spaces of a manifold, while the definition via velocities of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below, in the embedded manifold picture, a tangent vector at a point x is thought of as the velocity of a curve passing through the point x. We can therefore take a tangent vector to be a class of curves passing through x while being tangent to each other at x. Suppose M is a Ck manifold and x is a point in M. Pick a chart φ, U → Rn, where U is an open subset of M containing x. Suppose two curves γ1, → M and γ2, → M with γ1 = γ2 = x are given such that φ ∘ γ1, then γ1 and γ2 are called equivalent at 0 if the ordinary derivatives of φ ∘ γ1 and φ ∘ γ2 at 0 coincide. This defines a relation on such curves, and the equivalence classes are known as the tangent vectors of M at x
26.
Map (mathematics)
–
There are also a few, less common uses in logic and graph theory. In many branches of mathematics, the map is used to mean a function. For instance, a map is a function in topology. Some authors, such as Serge Lang, use only to refer to maps in which the codomain is a set of numbers. Sets of maps of special kinds are the subjects of many important theories, see for instance Lie group, mapping class group, in the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. A partial map is a function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, all these usages can be applied to maps as general functions or as functions with special properties. In category theory, map is used as a synonym for morphism or arrow. In formal logic, the map is sometimes used for a functional predicate. In graph theory, a map is a drawing of a graph on a surface without overlapping edges, if the surface is a plane then a map is a planar graph, similar to a political map
27.
Fraktur
–
Fraktur is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. From this, Fraktur is sometimes contrasted with the Latin alphabet in northern European texts, similarly, the term Fraktur or Gothic is sometimes applied to all of the blackletter typefaces. Besides the 26 letters of the Latin alphabet, Fraktur includes the ß, vowels with umlauts, and the ſ. Some Fraktur typefaces also include a variant form of the letter r known as the r rotunda, most older Fraktur typefaces make no distinction between the majuscules I and J, even though the minuscules i and j are differentiated. One difference between the Fraktur and other blackletter scripts is that in the lower case o, the part of the bow is broken. In Danish texts composed in Fraktur, the letter ø was already preferred to the German, Fraktur types for printing were established by the Augsburg publisher Johann Schönsperger at the issuance of a series of Maximilians works such as his Prayer Book or the illustrated Theuerdank poem. In the 18th century, the German Theuerdank Fraktur was further developed by the Leipzig typographer Johann Gottlob Immanuel Breitkopf to create the typeset Breitkopf Fraktur, while over the succeeding centuries, most Central Europeans switched to Antiqua, German-speakers remained a notable holdout. Some books at that time used related blackletter fonts such as Schwabacher, however, the predominant typeface was the Normalfraktur and this move was hotly debated in Germany, where it was known as the Antiqua–Fraktur dispute. The shift affected mostly scientific writing in Germany, whereas most belletristic literature, the press was scolded for its frequent use of Roman characters under Jewish influence and German émigrés were urged to use only German script. This radically changed on January 3,1941, when Martin Bormann issued a circular to all public offices which declared Fraktur to be Judenlettern, German historian Albert Kapr has speculated that the régime had realized that Fraktur would inhibit communication in the territories occupied during World War II. Even with the abolition of Fraktur, some publications include elements of it in headlines, very occasionally, academic works still used Fraktur in the text itself. Notably, Joachim Jeremiass work Die Briefe an Timotheus und Titus was published in 1963 using Fraktur, more often, some ligatures ch, ck from Fraktur were used in antiqua-typed editions. That continued mostly up to the offset type period, Fraktur saw a brief resurgence after the War, but quickly disappeared in a Germany keen on modernising its appearance. In this modern decorative use, the rules about the use of long s and short s. Individual Fraktur letters are used in mathematics, which often denotes associated or parallel concepts by the same letter in different fonts. For example, a Lie group is denoted by G. A ring ideal might be denoted by a while an element is a ∈ a, the Fraktur c is also used to denote the cardinality of the continuum, that is, the cardinality of the real line. In model theory, A is used to denote an arbitrary model, in Unicode, Fraktur is treated as a font of the Latin alphabet, and is not encoded separately
28.
Streamlines, streaklines, and pathlines
–
Streamlines, streaklines and pathlines are field lines in a fluid flow. They differ only when the changes with time, that is. These show the direction in which a fluid element will travel at any point in time. Streaklines are the loci of points of all the particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline, Pathlines are the trajectories that individual fluid particles follow. These can be thought of as recording the path of an element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time. Timelines are the lines formed by a set of particles that were marked at a previous instant in time. By definition, different streamlines at the instant in a flow do not intersect. However, pathlines are allowed to intersect themselves or other pathlines, Streamlines and timelines provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the full time-history of the flow. However, often sequences of timelines at different instants—being presented either in an image or with a video stream—may be used to provide insight in the flow. If a line, curve or closed curve is used as start point for a set of streamlines. In the case of a curve in a steady flow, fluid that is inside a stream surface must remain forever within that same stream surface. A scalar function whose contour lines define the streamlines is known as the stream function. Streamlines are defined by d x → S d s × u → =0, a streamtube consists of a bundle of streamlines, much like communication cable. Pathlines are defined by { d x → P d t = u → P x → P = x → P0 The suffix P indicates that we are following the motion of a fluid particle. Note that at point x → P the curve is parallel to the velocity vector u →. The parameter τ P, parametrizes the streakline x → P and 0 ≤ τ P ≤ t 0, in steady flow, the streamlines, pathlines, and streaklines coincide
29.
Wingtip vortices
–
Wingtip vortices are circular patterns of rotating air left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing, wingtip vortices are sometimes named trailing or lift-induced vortices because they also occur at points other than at the wing tips. Wingtip vortices are associated with induced drag, the imparting of downwash, careful selection of wing geometry, as well as of cruise conditions, are design and operational methods to minimize induced drag. Wingtip vortices form the primary component of wake turbulence, depending on ambient atmospheric humidity as well as the geometry and wing loading of aircraft, water may condense or freeze in the core of the vortices, making the vortices visible. When a wing generates aerodynamic lift the air on the top surface has lower pressure relative to the bottom surface, air flows from below the wing and out around the tip to the top of the wing in a circular fashion. An emergent circulatory flow pattern named vortex is observed, featuring a low-pressure core, three-dimensional lift and the occurrence of wingtip vortices can be approached with the concept of horseshoe vortex and described accurately with the Lanchester–Prandtl theory. In this view, the vortex is a continuation of the wing-bound vortex inherent to the lift generation. The result is a region of downwash behind the aircraft, between the two vortices, the two wingtip vortices do not merge because they are circulating in opposite directions. They dissipate slowly and linger in the atmosphere long after the airplane has passed and they are a hazard to other aircraft, known as wake turbulence. Wingtip vortices are associated with induced drag, a consequence of three-dimensional lift generation. The rotary motion of the air within the shed wingtip vortices reduces the angle of attack of the air on the wing. The lifting-line theory describes the shedding of trailing vortices as span-wise changes in lift distribution, for a given wing span and surface, minimal induced drag is obtained with an elliptical lift distribution. For a given distribution and surface, induced drag is reduced with increasing aspect ratio. As a consequence, aircraft for which a high ratio is desirable, such as gliders or long-range airliners. Another method of reducing induced drag is the use of winglets, winglets increase the effective aspect ratio of the wing, changing the pattern and magnitude of the vorticity in the vortex pattern. A reduction is achieved in the energy in the circular air flow. The cores of the vortices are sometimes visible because water present in them condenses from gas to liquid, the cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these regions form with little exchange of heat with the neighboring regions, so the local temperature in the low-pressure regions drops
30.
Atmospheric pressure
–
Atmospheric pressure, sometimes also called barometric pressure, is the pressure exerted by the weight of air in the atmosphere of Earth. In most circumstances atmospheric pressure is approximated by the hydrostatic pressure caused by the weight of air above the measurement point. As elevation increases, there is less overlying atmospheric mass, so that atmospheric pressure decreases with increasing elevation. On average, a column of air one square centimetre in cross-section, measured from sea level to the top of the atmosphere, has a mass of about 1.03 kilograms and that force is a pressure of 10.1 N/cm2 or 101 kN/m2. A column 1 square inch in cross-section would have a weight of about 14.7 lb or about 65.4 N and it is modified by the planetary rotation and local effects such as wind velocity, density variations due to temperature and variations in composition. The standard atmosphere is a unit of pressure defined as 101325 Pa, the mean sea level pressure is the average atmospheric pressure at sea level. This is the pressure normally given in weather reports on radio, television. When barometers in the home are set to match the weather reports, they measure pressure adjusted to sea level. The altimeter setting in aviation, is an atmospheric pressure adjustment, average sea-level pressure is 1013.25 mbar. In aviation weather reports, QNH is transmitted around the world in millibars or hectopascals, except in the United States, Canada, however, in Canadas public weather reports, sea level pressure is instead reported in kilopascals. The highest sea-level pressure on Earth occurs in Siberia, where the Siberian High often attains a sea-level pressure above 1050 mbar, the lowest measurable sea-level pressure is found at the centers of tropical cyclones and tornadoes, with a record low of 870 mbar. Pressure varies smoothly from the Earths surface to the top of the mesosphere, although the pressure changes with the weather, NASA has averaged the conditions for all parts of the earth year-round. As altitude increases, atmospheric pressure decreases, one can calculate the atmospheric pressure at a given altitude. Temperature and humidity affect the atmospheric pressure, and it is necessary to know these to compute an accurate figure. The graph at right was developed for a temperature of 15 °C, at low altitudes above the sea level, the pressure decreases by about 1.2 kPa for every 100 meters. See pressure system for the effects of air pressure variations on weather, Atmospheric pressure shows a diurnal or semidiurnal cycle caused by global atmospheric tides. This effect is strongest in tropical zones, with an amplitude of a few millibars and these variations have two superimposed cycles, a circadian cycle and semi-circadian cycle. The highest adjusted-to-sea level barometric pressure recorded on Earth was 1085.7 hPa measured in Tosontsengel
31.
Velocity
–
The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion, Velocity is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a vector quantity, both magnitude and direction are needed to define it. The scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI system as metres per second or as the SI base unit of. For example,5 metres per second is a scalar, whereas 5 metres per second east is a vector, if there is a change in speed, direction or both, then the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction, constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a path has a constant speed. Hence, the car is considered to be undergoing an acceleration, Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified, however, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be noticed when we consider movement around a circle and this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, average velocity can be calculated as, v ¯ = Δ x Δ t. The average velocity is less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, from this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity v is the displacement function x. In the figure, this corresponds to the area under the curve labeled s. Since the derivative of the position with respect to time gives the change in position divided by the change in time, although velocity is defined as the rate of change of position, it is often common to start with an expression for an objects acceleration. As seen by the three green tangent lines in the figure, an objects instantaneous acceleration at a point in time is the slope of the tangent to the curve of a v graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time, from there, we can obtain an expression for velocity as the area under an a acceleration vs. time graph
32.
Wind tunnel
–
A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. A wind tunnel consists of a passage with the object under test mounted in the middle. Air is made to move past the object by a fan system or other means. The test object, often called a wind tunnel model, is instrumented with sensors to measure aerodynamic forces, pressure distribution. The earliest wind tunnels were invented towards the end of the 19th century, in the days of aeronautic research. In that way an observer could study the flying object in action. The development of wind tunnels accompanied the development of the airplane, large wind tunnels were built during World War II. Wind tunnel testing was considered of importance during the Cold War development of supersonic aircraft. Determining such forces was required before building codes could specify the required strength of such buildings, in these studies, the interaction between the road and the vehicle plays a significant role, and this interaction must be taken into consideration when interpreting the test results. The advances in fluid dynamics modelling on high speed digital computers has reduced the demand for wind tunnel testing. However, CFD results are not completely reliable and wind tunnels are used to verify CFD predictions. Air velocity and pressures are measured in several ways in wind tunnels, air velocity through the test section is determined by Bernoullis principle. Measurement of the pressure, the static pressure, and the temperature rise in the airflow. The direction of airflow around a model can be determined by tufts of yarn attached to the aerodynamic surfaces, the direction of airflow approaching a surface can be visualized by mounting threads in the airflow ahead of and aft of the test model. Smoke or bubbles of liquid can be introduced into the upstream of the test model. Aerodynamic forces on the test model are usually measured with beam balances, connected to the test model with beams, strings, or cables. Pressure distributions can more conveniently be measured by the use of pressure-sensitive paint, the strip is attached to the aerodynamic surface with tape, and it sends signals depicting the pressure distribution along its surface. The aerodynamic properties of an object can not all remain the same for a scaled model, however, by observing certain similarity rules, a very satisfactory correspondence between the aerodynamic properties of a scaled model and a full-size object can be achieved
33.
Iron
–
Iron is a chemical element with symbol Fe and atomic number 26. It is a metal in the first transition series and it is by mass the most common element on Earth, forming much of Earths outer and inner core. It is the fourth most common element in the Earths crust, like the other group 8 elements, ruthenium and osmium, iron exists in a wide range of oxidation states, −2 to +6, although +2 and +3 are the most common. Elemental iron occurs in meteoroids and other low oxygen environments, but is reactive to oxygen, fresh iron surfaces appear lustrous silvery-gray, but oxidize in normal air to give hydrated iron oxides, commonly known as rust. Unlike the metals that form passivating oxide layers, iron oxides occupy more volume than the metal and thus flake off, Iron metal has been used since ancient times, although copper alloys, which have lower melting temperatures, were used even earlier in human history. Pure iron is soft, but is unobtainable by smelting because it is significantly hardened and strengthened by impurities, in particular carbon. A certain proportion of carbon steel, which may be up to 1000 times harder than pure iron. Crude iron metal is produced in blast furnaces, where ore is reduced by coke to pig iron, further refinement with oxygen reduces the carbon content to the correct proportion to make steel. Steels and iron alloys formed with metals are by far the most common industrial metals because they have a great range of desirable properties. Iron chemical compounds have many uses, Iron oxide mixed with aluminium powder can be ignited to create a thermite reaction, used in welding and purifying ores. Iron forms binary compounds with the halogens and the chalcogens, among its organometallic compounds is ferrocene, the first sandwich compound discovered. Iron plays an important role in biology, forming complexes with oxygen in hemoglobin and myoglobin. Iron is also the metal at the site of many important redox enzymes dealing with cellular respiration and oxidation and reduction in plants. A human male of average height has about 4 grams of iron in his body and this iron is distributed throughout the body in hemoglobin, tissues, muscles, bone marrow, blood proteins, enzymes, ferritin, hemosiderin, and transport in plasma. The mechanical properties of iron and its alloys can be evaluated using a variety of tests, including the Brinell test, Rockwell test, the data on iron is so consistent that it is often used to calibrate measurements or to compare tests. An increase in the content will cause a significant increase in the hardness. Maximum hardness of 65 Rc is achieved with a 0. 6% carbon content, because of the softness of iron, it is much easier to work with than its heavier congeners ruthenium and osmium. Because of its significance for planetary cores, the properties of iron at high pressures and temperatures have also been studied extensively
34.
Maxwell's equations
–
Maxwells equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio such as power generation, electric motors, wireless communication, cameras, televisions. Maxwells equations describe how electric and magnetic fields are generated by charges, currents, one important consequence of the equations is the demonstration of how fluctuating electric and magnetic fields can propagate at the speed of light. This electromagnetic radiation manifests itself in ways from radio waves to light. The equations have two major variants, the microscopic Maxwell equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges, the macroscopic Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. However, their use requires experimentally determining parameters for a description of the electromagnetic response of materials. The term Maxwells equations is used for equivalent alternative formulations. The space-time formulations, are used in high energy and gravitational physics because they make the compatibility of the equations with special and general relativity manifest. In many situations, though, deviations from Maxwells equations are immeasurably small, exceptions include nonclassical light, photon-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons. In the electric and magnetic field there are four equations. The two inhomogeneous equations describe how the fields vary in space due to sources, Gausss law describes how electric fields emanate from electric charges. Gausss law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles, the two homogeneous equations describe how the fields circulate around their respective sources. A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles, a version of this law was included in the original equations by Maxwell but, by convention, is no longer. The precise formulation of Maxwells equations depends on the definition of the quantities involved. Conventions differ with the systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of light c. This makes constants come out differently, the vector calculus formulation below has become standard. For formulations using tensor calculus or differential forms, see alternative formulations, for relativistically invariant formulations, see relativistic formulations
35.
Gradient
–
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
36.
Del
–
Del, or nabla, is an operator used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a 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
37.
Conservative vector field
–
In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential. Conservative vector fields have the property that the integral is path independent. Path independence of the integral is equivalent to the vector field being conservative. A conservative vector field is irrotational, in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected, conservative vector fields appear naturally in mechanics, They are vector fields representing forces of physical systems in which energy is conserved. Therefore, in general, the value of the integral depends on the path taken, although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy. This is because a field is conservative. As an example of a field, imagine pushing a box from one end of a room to another. Pushing the box in a line across the room requires noticeably less work against friction than along a curved path covering a greater distance. It is rotational in that one can keep getting higher or keep getting lower while going around in circles and it is non-conservative in that one can return to ones starting point while ascending more than one descends or vice versa. On a real staircase, the height above the ground is a potential field, If one returns to the same place. Its gradient would be a vector field and is irrotational. The situation depicted in the painting is impossible, here, ∇ φ denotes the gradient of φ. When the equation holds, φ is called a scalar potential for v. The Fundamental Theorem of Vector Calculus states that any field can be expressed as the sum of a conservative vector field. A key property of a vector field v is that its integral along a path depends only on the endpoints of that path. Suppose that P is a path in U with initial point A. If v = ∇ φ for some C1 scalar field φ so that v is a vector field
38.
Gradient descent
–
Gradient descent is a first-order iterative optimization algorithm. To find a minimum of a function using gradient descent. If instead one takes steps proportional to the positive of the gradient, one approaches a maximum of that function. Gradient descent is known as steepest descent, or the method of steepest descent. Gradient descent should not be confused with the method of steepest descent for approximating integrals and it follows that, if a n +1 = a n − γ ∇ F for γ small enough, then F ≥ F. In other words, the term γ ∇ F is subtracted from a because we want to move against the gradient and we have F ≥ F ≥ F ≥ ⋯, so hopefully the sequence converges to the desired local minimum. Note that the value of the step size γ is allowed to change at every iteration. With certain assumptions on the function F and particular choices of γ, γ n = T | | ∇ F − ∇ F | |2 convergence to a minimum can be guaranteed. When the function F is convex, all local minima are also global minima and this process is illustrated in the adjacent picture. Here F is assumed to be defined on the plane, the blue curves are the contour lines, that is, the regions on which the value of F is constant. A red arrow originating at a point shows the direction of the gradient at that point. Note that the gradient at a point is orthogonal to the line going through that point. We see that gradient descent leads us to the bottom of the bowl, Gradient descent has problems with pathological functions such as the Rosenbrock function shown here. The Rosenbrock function has a curved valley which contains the minimum. The bottom of the valley is very flat, because of the curved flat valley the optimization is zig-zagging slowly with small stepsizes towards the minimum. The Zig-Zagging nature of the method is also evident below, where the gradient descent method is applied to F = sin cos . For some of the examples, gradient descent is relatively slow close to the minimum, technically. For poorly conditioned convex problems, gradient descent increasingly zigzags as the point nearly orthogonally to the shortest direction to a minimum point
39.
Orthogonal group
–
Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation