Classical treatment of tensors

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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:

$T^{\left[i_1,i_2,i_3,...i_n\right]}_{\left[j_1,j_2,j_3,...j_m\right]}$

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.

1 Contravariant and covariant tensors
2 General tensors
3 See also
4 Further reading

[edit] Contravariant and covariant tensors

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

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

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

$\bar{T}_i = T_r\frac{\partial x^r}{\partial \bar{x}^i}.$

[edit] General tensors

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

$T^{\left[i_1,i_2,...i_p\right]}_{\left[j_1,j_2,...j_q\right]} = T^{i_1} \otimes T^{i_2} ... \otimes T^{i_p} \otimes T_{j_1} \otimes T_{j_2} ... \otimes T_{j_q}$

such that:

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