Tensor

An $n$th-Rank tensor of order $m$ is a mathematical object in $m$-dimensional space which has $n$ indices and $m^n$ components and obeys certain transformation rules. Each index of a tensor ranges over the number of dimensions of Space. If the components of any tensor of any Rank vanish in one particular coordinate system, they vanish in all coordinate systems.

Zeroth-Rank tensors are called Scalars, and first-Rank tensors are called Vectors. In tensor notation, a vector v would be written $v_i$, where $i=1$, ..., $m$. Tensor notation can provide a very concise way of writing vector and more general identities. For example, in tensor notation, the Dot Product ${\bf u}\cdot {\bf v}$ is simply written

$${\bf u}\cdot{\bf v}=u_iv_i,$$ (1)

where repeated indices are summed over (Einstein Summation) so that $u_iv_i$ stands for $u_1v_1+\ldots+u_mv_m$. Similarly, the Cross Product can be concisely written as

$${\bf u}\times{\bf v}=\epsilon_{ijk}u^jv^k,$$ (2)

where $\epsilon_{ijk}$ is the Levi-Civita Tensor.

Second-Rank tensors resemble square Matrices. Contravariant second-Rank tensors are objects which transform as

$$A'^{ij} = {\partial x_i'\over\partial x_k} {\partial x_j'\over\partial x_l}A^{kl}.$$ (3)

Covariant second-Rank tensors are objects which transform as

$$C'_{ij} = {\partial x_k\over\partial x_i'} {\partial x_l\over\partial x_j'}C_{kl}.$$ (4)

Mixed second-Rank tensors are objects which transform as

$${B'}_j^i = {\partial x_i'\over\partial x_k} {\partial x_l\over\partial x_j'}B^k_l.$$ (5)

If two tensors $A$ and $B$ have the same Rank and the same Covariant and Contravariant indices, then

$$A^{ij}+B^{ij} = C^{ij}$$ (6)

$$A_{ij}+B_{ij} = C_{ij}$$ (7)

$$A^i_j+B^i_j = C^i_j.$$ (8)

A transformation of the variables of a tensor changes the tensor into another whose components are linear Homogeneous Functions of the components of the original tensor.

See also Antisymmetric Tensor, Curl, Divergence, Gradient, Irreducible Tensor, Isotropic Tensor, Jacobi Tensor, Ricci Tensor, Riemann Tensor, Scalar, Symmetric Tensor, Torsion Tensor, Vector, Weyl Tensor

References

Tensors

Abraham, R.; Marsden, J. E.; and Ratiu, T. S. Manifolds, Tensor Analysis, and Applications. New York: Springer-Verlag, 1991.

Akivis, M. A. and Goldberg, V. V. An Introduction to Linear Algebra and Tensors. New York: Dover, 1972.

Arfken, G. ``Tensor Analysis.'' Ch. 3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 118-167, 1985.

Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, 1989.

Bishop, R. and Goldberg, S. Tensor Analysis on Manifolds. New York: Dover, 1980.

Jeffreys, H. Cartesian Tensors. Cambridge, England: Cambridge University Press, 1931.

Joshi, A. W. Matrices and Tensors in Physics, 3rd ed. New York: Wiley, 1995.

Lass, H. Vector and Tensor Analysis. New York: McGraw-Hill, 1950.

Lawden, D. F. An Introduction to Tensor Calculus, Relativity, and Cosmology, 3rd ed. Chichester, England: Wiley, 1982.

McConnell, A. J. Applications of Tensor Analysis. New York: Dover, 1947.

Morse, P. M. and Feshbach, H. ``Vector and Tensor Formalism.'' §1.5 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44-54, 1953.

Simmonds, J. G. A Brief on Tensor Analysis, 2nd ed. New York: Springer-Verlag, 1994.

Sokolnikoff, I. S. Tensor Analysis--Theory and Applications, 2nd ed. New York: Wiley, 1964.

Synge, J. L. and Schild, A. Tensor Calculus. New York: Dover, 1978.

Wrede, R. C. Introduction to Vector and Tensor Analysis. New York: Wiley, 1963.