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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 a system of quantities that satisfies a multi-dimensional transformation law when passing from one coordinate system to another. It takes the form:

$T^{i_1,i_2,i_3,...i_n}_{j_1,j_2,j_3,...j_m}$

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.

[edit] 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

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

[edit] 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

$T^{i_1,i_2,\dots,i_p}_{j_1,j_2,\dots,j_q} = T^{i_1} \otimes T^{i_2} \otimes\cdots\otimes T^{i_p} \otimes T_{j_1} \otimes T_{j_2} \otimes\cdots\otimes T_{j_q}$

such that:

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

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.

[edit] 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

[edit] 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.

Contents

[edit] Contravariant and covariant vectors

[edit] General tensors

[edit] See also

[edit] References