Classical treatment of tensors

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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.

The Einstein notation is used throughout this page. For help with notation, 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, that takes forms in one coordinate system into another. It takes the form:

<math>T^{\left[i_1,i_2,i_3,...i_n\right]}_{\left[j_1,j_2,j_3,...j_n\right]}.<math>

The new coordinate system is represented by being 'barred'(<math>\bar{x}^i<math>), and the old coordinate system is unbarred(<math>x^i<math>).

The upper indices [<math>i_1,i_2,i_3,...i_n<math>] are the contravariant components, and the lower indices [<math>j_1,j_2,j_3,...j_n<math>] are the covariant components.

Contents

Contravariant and covariant tensors

A contravariant tensor of order 1(<math>T^i<math>) is defined as:

<math>\bar{T}^i = T^r\frac{\partial \bar{x}^i}{\partial x^r}.<math>

A covariant tensor of order 1(<math>T_i<math>) is defined as:

<math>\bar{T}_i = T_r\frac{\partial x^r}{\partial \bar{x}^i}.<math>

General tensors

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

<math>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_p}<math>

such that:

<math>\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}}. <math>

This is sometimes termed the tensor transformation law.

More about tensors

Further reading

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