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Undid revision 1296100551 by TASPlasma (talk) seems like this decreased rather than increased consistency |
Undid revision 1296253827 by JayBeeEll (talk) never mind, I see it's locally consistent with this paragraph and globally inconsistent in the article |
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{{Main|Multilinear map}}
A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in [[differential geometry]] is to define tensors relative to a fixed (finite-dimensional) vector space ''V'', which is usually taken to be a particular vector space of some geometrical significance like the [[tangent space]] to a manifold.<ref>{{citation|last=Lee|first=John|title=Introduction to smooth manifolds|url={{google books |plainurl=y |id=4sGuQgAACAAJ|page=173}}|page=173|year=2000|publisher=Springer|isbn=978-0-387-95495-0}}</ref> In this approach, a type {{nowrap|(''p'', ''q'')}} tensor ''T'' is defined as a [[multilinear map]],
:<math> T: \underbrace{V^* \times\dots\times V^*}_{p \text{ copies}} \times \underbrace{ V \times\dots\times V}_{q \text{ copies}} \rightarrow \
where ''V''<sup>∗</sup> is the corresponding [[dual space]] of covectors, which is linear in each of its arguments. The above assumes ''V'' is a vector space over the [[real number]]s, {{tmath|\R}}. More generally, ''V'' can be taken over any [[Field (mathematics)|field]] ''F'' (e.g. the [[complex number]]s), with ''F'' replacing {{tmath|\R}} as the codomain of the multilinear maps.
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