Tensor

An (N, M)-tensor at a given point in space can be described by a set of numbers with N + M indices which transforms, upon coordinate transformation given by the matrix A, in the following way:
An (N, M)-tensor in a three-dimensional manifold therefore has 3 components. It is contravariant in N components and covariant in M components. Tensors of the type:-
 t_{\alpha^\beta \gamma} 
-are of course not excluded, as they can be constructed from the above kind of tensors
by rearrangement of indices. Matrices (2 indices), vectors and covectors (1 index) and scalars (0 indices) are therefore also tensors, where the latter transforms as s′ = s.
Each index of a tensor ranges over the number of dimensions of space. However, the dimension of the space is largely irrelevant in most tensor equations (with the notable exception of the contracted Kronecker delta). Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.
Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity.
Category:Physics