Classical treatment of tensors

A tensor is a generalization of the concepts of vectors and matrices. Tensors allow one to express physical laws in a form that applies to any coordinate system. For this reason, they are used extensively in continuum mechanics and the theory of relativity.

A tensor is an invariant multi-dimensional transformation, one that takes forms in one coordinate system into another. It takes the form:

:$T^\{left\; [i\_1,i\_2,i\_3,...i\_n\; ight]\; \}\_\{left\; [j\_1,j\_2,j\_3,...j\_m\; ight]\; \}$

The new coordinate system is represented by being 'barred'(), and the old coordinate system is unbarred($x^i$).

The upper indices [$i\_1,i\_2,i\_3,...i\_n$] are the contravariant components, and the lower indices [$j\_1,j\_2,j\_3,...j\_n$] are the covariant components.

Contravariant and covariant tensors

A contravariant tensor of order 1($T^i$) is defined as:

:

A covariant tensor of order 1($T\_i$) is defined as:

:

General tensors

A multi-order (general) tensor is simply the tensor product of single order tensors:

:$T^\{left\; [i\_1,i\_2,...i\_p\; ight]\; \}\_\{left\; [j\_1,j\_2,...j\_q\; ight]\; \}\; =\; T^\{i\_1\}\; otimes\; T^\{i\_2\}\; ...\; otimes\; T^\{i\_p\}\; otimes\; T\_\{j\_1\}\; otimes\; T\_\{j\_2\}\; ...\; otimes\; T\_\{j\_q\}$

such that:

:

This is sometimes termed the tensor transformation law.

See also

* Tensor derivative
* Absolute differentiation
* Curvature
* Riemannian geometry

Further reading

* Schaum's Outline of Tensor Calculus
* Synge and Schild, Tensor Calculus, Toronto Press: Toronto, 1949