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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\right]}_{\left[j_1,j_2,j_3,...j_m\right]}](http://upload.wikimedia.org/math/3/5/4/354e8cc202d5ff96f2111e178d0fb5a8.png)
The new coordinate system is represented by being 'barred'( ), and the old coordinate system is unbarred(xi).
The upper indices [i1,i2,i3,...in] are the contravariant components, and the lower indices [j1,j2,j3,...jn] are the covariant components.
Contravariant and covariant tensors
A contravariant tensor of order 1(Ti) is defined as:
A covariant tensor of order 1(Ti) is defined as:
General tensors
A multi-order (general) tensor is simply the tensor product of single order tensors:
![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_q}](http://upload.wikimedia.org/math/7/c/0/7c0bbe8abd9846eebaa949d5d60c53ce.png)
such that:
![\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}}.](http://upload.wikimedia.org/math/a/6/f/a6f3ca7f7686a6c79ca68b6cba857169.png)
This is sometimes termed the tensor transformation law.
See also
Further reading
- Schaum's Outline of Tensor Calculus
- Synge and Schild, Tensor Calculus, Toronto Press: Toronto, 1949
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