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The Einstein summation convention is used throughout this page. For help with notation, refer to the table of mathematical symbols.
A tensor is a generalization of the concept of vector 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 multidimensional transformation, that takes forms in one coordinate system into another. It takes the form:
The new coordinate system is represented by being 'barred'(<math>\bar{x}^i</math>), and the old coordinate system is unbarred(<math>x^i</math>).
The upper indices [<math>i_1,i_2,i_3,...i_n</math>] are the contravariant components, and the lower indices [<math>j_1,j_2,j_3,...j_n</math>] are the covariant components.

Contravariant and covariant tensors A contravariant tensor of order 1(<math>T^i</math>) is defined as:
A covariant tensor of order 1(<math>T_i</math>) is defined as:
General tensors A multiorder (general) tensor is simply the tensor product of single order tensors:
such that:
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