Tensor: Difference between revisions

Revision as of 22:57, 23 September 2024

Revision as of 00:23, 24 September 2024

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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.

In viewing a tensor as a multilinear map, it is conventional to identify the [[double dual]] ''V''^∗∗ of the vector space ''V'', that isi.e., the space of linear functionals on the dual vector space ''V''^∗, with the vector space ''V''. There is always a [[Dual space#Injection into the double-dual|natural linear map]] from ''V'' to its double dual, given by evaluating a linear form in ''V''^∗ against a vector in ''V''. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify ''V'' with its double dual.

=== Using tensor products ===

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:<math>T \in \underbrace{V \otimes\dots\otimes V}_{p\text{ copies}} \otimes \underbrace{V^* \otimes\dots\otimes V^*}_{q \text{ copies}}.</math>

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>}}, that is,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>

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| 3-form E.g. [[multipole moment|octupole moment]]

|

| E.g. ''M''-form, that is,i.e. [[volume form]]

|

|-

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{{Main|Tensor product}}

The [[tensor product]] takes two tensors, ''S'' and ''T'', and produces a new tensor, {{nowrap|{{math|''S'' ⊗ ''T''}}}}, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, that isi.e.,

<math display="block">(S \otimes T)(v_1, \ldots, v_n, v_{n+1}, \ldots, v_{n+m}) = S(v_1, \ldots, v_n)T(v_{n+1}, \ldots, v_{n+m}),</math>

which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, that isi.e.,

<math display="block">

(S \otimes T)^{i_1\ldots i_l i_{l+1}\ldots i_{l+n}}_{j_1\ldots j_k j_{k+1}\ldots j_{k+m}} =