Revision as of 23:35, 19 September 2024 edit Æolus (talk \| contribs) 53 edits →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 ← Previous edit		Revision as of 00:31, 20 September 2024 edit undo Tito Omburo (talk \| contribs) Extended confirmed users 3,069 edits →As multilinear maps: restored notation Next edit →
Line 93: 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.

Tensor: Difference between revisions