Tensor: Difference between revisions

Revision as of 00:31, 20 September 2024

Revision as of 00:33, 20 September 2024

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A basis {{math|''v''_''i''}} of {{math|''V''}} and basis {{math|''w''_''j''}} of {{math|''W''}} naturally induce a basis {{math|''v''_''i'' ⊗ ''w''_''j''}} of the tensor product {{math|''V'' ⊗ ''W''}}. The components of a tensor {{math|''T''}} are the coefficients of the tensor with respect to the basis obtained from a basis {{math|<nowiki>{</nowiki>'''e'''_''i''<nowiki>}</nowiki>}} for {{math|''V''}} and its dual basis {{math|{'''''ε'''''{{i sup|''j''}}<nowiki>}</nowiki>}}, i.e.

:<math>T = {T^{i_1\dots i_p}}\!_{j_1\dots j_q}\; \mathbf{e}_{i_1}\otimes\cdots\otimes \mathbf{e}_{i_p}\otimes \boldsymbol{\varepsilon}^{j_1}\otimes\cdots\otimes \boldsymbol{\varepsilon}^{j_q}.</math>

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type {{math|(''p'', ''q'')}} tensor. Moreover, the universal property of the tensor product gives a [[bijection|one-to-one correspondence]] between tensors defined in this way and tensors defined as multilinear maps.