Classical treatment of tensors

Disambiguation

Note: The following is a component-based "classical" treatment of tensors. See Component-free treatment of tensors for a modern abstract treatment, and Intermediate treatment of tensors for an approach which bridges the two. This article uses Einstein notation. For help, refer to the table of mathematical symbols.

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:

$T^\{left[i\_1,i\_2,i\_3,...i\_nright]\}\_\{left[j\_1,j\_2,j\_3,...j\_mright]\}$

The new coordinate system is represented by being 'barred'($bar\{x\}^i$), 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:

$bar\{T\}^i\; =\; T^rfrac\{partial\; bar\{x\}^i\}\{partial\; x^r\}.$

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

$bar\{T\}\_i\; =\; T\_rfrac\{partial\; x^r\}\{partial\; bar\{x\}^i\}.$

General tensors

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

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

such that:

$bar\{T\}^\{left[i\_1,i\_2,...i\_pright]\}\_\{left[j\_1,j\_2,...j\_qright]\}\; =$

T^{left[r_1,r_2,...r_pright]}_{left[s_1,s_2,...s_qright]} frac{partial bar{x}^{i_1}}{partial x^{r_1}} frac{partial bar{x}^{i_2}}{partial x^{r_2}} ... frac{partial bar{x}^{i_p}}{partial x^{r_p}} frac{partial x^{s_1}}{partial bar{x}^{j_1}} frac{partial x^{s_2}}{partial bar{x}^{j_2}} ... frac{partial x^{s_q}}{partial bar{x}^{j_q}}.

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