Classical treatment of tensors

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 ight] }_{left [j_1,j_2,j_3,...j_m ight] }

The new coordinate system is represented by being 'barred'($$ar{x}^i), and the old coordinate system is unbarred($$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.

Contravariant and covariant tensors

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

:$$ar{T}^i = T^rfrac{partial ar{x}^i}{partial x^r}.

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

:$$ar{T}_i = T_rfrac{partial x^r}{partial ar{x}^i}.

General tensors

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

:$$T^{left [i_1,i_2,...i_p ight] }_{left [j_1,j_2,...j_q ight] } = 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:

:$$ar{T}^{left [i_1,i_2,...i_p ight] }_{left [j_1,j_2,...j_q ight] } = T^{left [r_1,r_2,...r_p ight] }_{left [s_1,s_2,...s_q ight] }frac{partial ar{x}^{i_1{partial x^{r_1frac{partial ar{x}^{i_2{partial x^{r_2...frac{partial ar{x}^{i_p{partial x^{r_pfrac{partial x^{s_1{partial ar{x}^{j_1frac{partial x^{s_2{partial ar{x}^{j_2...frac{partial x^{s_q{partial ar{x}^{j_q.

This is sometimes termed the tensor transformation law.

See also

* Tensor derivative
* Absolute differentiation
* Curvature
* Riemannian geometry

Further reading

* Schaum's Outline of Tensor Calculus
* Synge and Schild, Tensor Calculus, Toronto Press: Toronto, 1949