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→Definition: Where tensor components are shown in direct juxtaposition with the (co)basis elements, clarified ordering of component indices. (This is useful e.g. in General Relativity, where can switch to the dual basis on a single index). Also fixed typo 'archived'. Tags: Mobile edit Mobile app edit Android app edit App section source |
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then the multidimensional array obeys the transformation law
:<math>
</math> <math>
T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}]
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By applying a multilinear map ''T'' of type {{nowrap|(''p'', ''q'')}} to a basis {'''e'''<sub>''j''</sub>} for ''V'' and a canonical cobasis {'''ε'''<sup>''i''</sup>} for ''V''<sup>∗</sup>,
:<math>{T^{i_1\dots i_p}}\!_{j_1\dots j_q} \equiv T\left(\boldsymbol{\varepsilon}^{i_1}, \ldots,\boldsymbol{\varepsilon}^{i_p}, \mathbf{e}_{j_1}, \ldots, \mathbf{e}_{j_q}\right),</math>
a {{nowrap|(''p'' + ''q'')}}-dimensional array of components can be obtained. A different choice of basis will yield different components. But, because ''T'' is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of ''T'' thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map ''T''. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.
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A basis {{math|''v''<sub>''i''</sub>}} of {{math|''V''}} and basis {{math|''w''<sub>''j''</sub>}} of {{math|''W''}} naturally induce a basis {{math|''v''<sub>''i''</sub> ⊗ ''w''<sub>''j''</sub>}} 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'''<sub>''i''</sub><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.
This 1 to 1 correspondence can be
:<math>U \otimes V \cong\left(U^{* *}\right) \otimes\left(V^{* *}\right) \cong\left(U^{*} \otimes V^{*}\right)^{*} \cong \operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right)</math>
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