If A(r, s, t) be a function of the coordinates in n-dimensional space such that for an arbitrary vector B' of the type indicated by the index a A(r. s, t)B' is equal to the component C" of a contravariant tensor of order two. Prove that A(r, s, t) are the components of a tensor of the form A.
Answers
Explanation:
In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
The second-order Cauchy stress tensor ({\displaystyle \mathbf {T} }{\mathbf {T}}) describes the stress forces experienced by a material at a given point. The product {\displaystyle \mathbf {T} \cdot \mathbf {v} }{\displaystyle \mathbf {T} \cdot \mathbf {v} } of the stress tensor and a unit vector {\displaystyle \mathbf {v} }\mathbf {v} , pointing in a given direction, is a vector describing the stress forces experienced by a material at the point described by the stress tensor, along a plane perpendicular to {\displaystyle \mathbf {v} }\mathbf {v} . This image shows the stress vectors along three perpendicular directions, each represented by a face of the cube. Since the stress tensor describes a mapping that takes one vector as input, and gives one vector as output, it is a second-order tensor.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 - continuing the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others - as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1]