Classical treatment of tensors

Disambiguation

Note: The following is a component-based "classical" treatment of tensors. See Tensor (intrinsic definition) 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 a system of quantities that satisfies a multi-dimensional transformation law when passing from one coordinate system to another. It takes the form:

$T_{j_{1},j_{2},j_{3},...j_{m}}^{i_{1},i_{2},i_{3},...i_{n}}$

Notice that the contravariant indices $i_{1},i_{2},i_{3},...i_{n}$ are written as superscripts, and that the covariant indices $j_{1},j_{2},j_{3},...j_{m}$ are written as subscripts.

Contravariant and covariant vectors

This section makes use of the Einstein notation convention, to represent tensors. When transforming coordinates, quantities in the new coordinate system are represented by being 'barred'(${\bar {x}}^{i}$), and quantities in the old coordinate system are unbarred($x^{i}$).

A contravariant vector is a tensor of order 1($T^{i}$) which transforms as

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

at each event in spacetime when one goes from the $x\,$ coordinate system to the ${\bar {x}}\,$ coordinate system.

A covariant vector is a tensor of order 1($T_{i}$) which transforms as

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

General tensors

In general, the value of a tensor field at an event in spacetime is an element of a vector space which is the tensor product of multiple copies of the tangent space (contravariant vectors) and multiple copies of the cotangent space (covariant vectors). As such, it is a smooth (C^∞) mapping from the base space of a vector bundle to the total space which when projected back onto the base space has returned to its starting point.

The tensor product of contravariant and covariant vectors is a tensor

$T_{j_{1},j_{2},\dots ,j_{q}}^{i_{1},i_{2},\dots ,i_{p}}=T^{i_{1}}\otimes T^{i_{2}}\otimes \cdots \otimes T^{i_{p}}\otimes T_{j_{1}}\otimes T_{j_{2}}\otimes \cdots \otimes T_{j_{q}}$

such that:

${\bar {T}}_{j_{1},j_{2},\dots ,j_{q}}^{i_{1},i_{2},\dots ,i_{p}}=T_{s_{1},s_{2},\dots ,s_{q}}^{r_{1},r_{2},\dots ,r_{p}}{\frac {\partial {\bar {x}}^{i_{1}}}{\partial x^{r_{1}}}}{\frac {\partial {\bar {x}}^{i_{2}}}{\partial x^{r_{2}}}}\cdots {\frac {\partial {\bar {x}}^{i_{p}}}{\partial x^{r_{p}}}}{\frac {\partial x^{s_{1}}}{\partial {\bar {x}}^{j_{1}}}}{\frac {\partial x^{s_{2}}}{\partial {\bar {x}}^{j_{2}}}}\cdots {\frac {\partial x^{s_{q}}}{\partial {\bar {x}}^{j_{q}}}}.$

This is sometimes termed the tensor transformation law. The sum of two tensors satisfying the same transformation law also satisfies it, and is thus called a tensor. The difference of two tensors satisfying the same transformation law also satisfies it. The product of a tensor and a real number is a tensor satisfying the same transformation law. This linearity & homogenity, makes the space of tensors itself a vector space.

See also

• Tensor density, a generalization of tensors.
• Covariant derivative, a tensor derivative which treats the metric as constant.
• Lie derivative, a tensor derivative which treats a given contravariant vector as constant.
• Curvature
• Riemannian geometry

References

• Kay, David C (1988-04-01). Schaum's Outline of Tensor Calculus. McGraw-Hill. ISBN 978-0070334847.
• Synge JL, Schild A (1978-07-01). Tensor Calculus. Dover Publications. ISBN 978-0486636122.