Volume

Volume - Wikipedia: "Integrating the volume form gives the volume of the manifold according to that form."



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Volume form - Wikipedia

Volume form - Wikipedia:



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Multilinear form - Wikipedia

Multilinear form - Wikipedia: "Given a basis {\displaystyle (v_{1},\ldots ,v_{n})} for {\displaystyle V} and its dual {\displaystyle (\phi ^{1},\ldots ,\phi ^{n})} for dual vector space {\displaystyle V^{*}={\mathcal {A}}_{1}(V)} , the wedge products {\displaystyle \phi ^{i_{1}}\wedge \cdots \wedge \phi ^{i_{k}}} , with {\displaystyle 1\leq i_{1}<\cdots

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One-form - Wikipedia

One-form - Wikipedia: "Gravitation"



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Linear map - Wikipedia

Linear map - Wikipedia: "Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V).
If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.
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Linear map - Wikipedia

Linear map - Wikipedia: "If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.
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Generalized function - Wikipedia

Generalized function - Wikipedia: "A further way in which the theory has been extended is as generalized sections of a smooth vector bundle. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have compact support. The most developed theory is that of De Rham currents, dual to differential forms. These are homological in nature, in the way that differential forms give rise to De Rham cohomology. They can be used to formulate a very general Stokes' theorem."



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