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Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies :T_ = T_. The space of symmetric tensors of order ''r'' on a finite-dimensional vector space ''V'' is naturally isomorphic to the dual of the space of homogeneous polynomials of degree ''r'' on ''V''. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on ''V''. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics. Definition Let ''V'' be a vector space and :T\in V^ a tensor of order ''k''. Then ''T'' is a symmetric tensor if :\tau_\sigma T = T\, for the braiding maps associated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Bilinear Form
In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B that maps every pair (u,v) of elements of the vector space V to the underlying field such that B(u,v)=B(v,u) for every u and v in V. They are also referred to more briefly as just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for ''V''. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given a symmetric bilinear form ''B'', the function is the associated quadratic form on the vector space. Moreover, if the characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondegenerate Form
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definition when ''V'' is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero ''x'' in ''V'' such that :f(x,y)=0\, for all \,y \in V. Nondegenerate forms A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if :f(x,y)=0 for all y \in V implies that x = 0. The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map V \to V^* be an isomorphism, not positivity. For example, a manifold with an inner product structure on its t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate-free
A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system. Benefits Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater mathematical elegance at the cost of some abstraction from the detailed formulae needed to evaluate these equations within a particular system of coordinates. In addition to elegance, coordinate-free treatments are crucial in certain applications for proving that a given definition is well formulated. For example, for a vector space V with basis v_1, ..., v_n, it may be tempting to construct the dual space V^* as the formal span of the symbols v_1^*, ..., v_n^* with bracket \langle \sum \alpha_i v_i , \sum \beta_i v_i^* \rangle := \sum \alpha_i \beta_i, but it is not immediately clear that this construction is independent of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Contravariance Of Vectors
In physics, especially 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 modern mathematical notation, the role is sometimes swapped. 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, by changing scale from meters to centimeters (that is, ''dividing'' the scale of the reference axes by 100), the components of a measured velocity vector are ''multiplied'' by 100. A vector changes scale ''inversely'' to changes in scale to the reference axes, and consequently is called ''contravariant''. As a result, a vector often has units of distance or distance with other units (as, for example, velocity has units of distance divided by time). In contrast, a covector, also called a ''dual vector'', typically has units of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Basis
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as :\mathbf_ = \lim_ \frac , where is the displacement vector between the point and a nearby point whose coordinate separation from is along the coordinate curve (i.e. the curve on the manifold through for which the local coordinate varies and all other coordinates are constant). It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve on the manifold defined by with the tangent vector , where , and a function defined in a neighbourhood of , the variation of along can be written as :\frac = \frac\frac = u^ \frac f . Since we have that , the identification is often made between a coordinate basis vector and the partial derivative operator , under the interpretation of vectors as operators acting on functions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on . Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor ''field'' as it is defined on a manifold: it is named after Bernhard Riemann, and associates a tensor to each point of a Riemannia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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