Editing Differential form (section)

=== Multi-index notation ===
A common notation for the wedge product of elementary {{mvar|k}}-forms is so called [[multi-index notation]]: in an {{mvar|n}}-dimensional context, for {{nowrap|<math>I = (i_1, i_2,\ldots , i_k), 1 \leq i_1 < i_2 < \cdots < i_k \leq n</math>,}} we define {{nowrap|<math display="inline">dx^I := dx^{i_1} \wedge \cdots \wedge dx^{i_k} = \bigwedge_{i\in I} dx^i</math>.}}<ref>{{Cite book|title=An introduction to manifolds|last=Tu, Loring W.|date=2011|publisher=Springer|isbn=9781441974006|edition= 2nd|location=New York|oclc=682907530}}</ref> Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length {{mvar|k}}, in a space of dimension {{mvar|n}}, denoted {{nowrap|<math>\mathcal{J}_{k,n} := \{I=(i_1,\ldots,i_k):1\leq i_1<i_2<\cdots<i_k\leq n\}</math>.}} Then locally (wherever the coordinates apply), <math>\{dx^I\}_{I \in \mathcal{J}_{k,n}}</math> spans the space of differential {{mvar|k}}-forms in a manifold {{mvar|M}} of dimension {{mvar|n}}, when viewed as a module over the ring {{math|''C''{{sup|∞}}(''M'')}} of smooth functions on {{mvar|M}}. By calculating the size of <math>\mathcal{J}_{k,n}</math> combinatorially, the module of {{mvar|k}}-forms on an {{mvar|n}}-dimensional manifold, and in general space of {{mvar|k}}-covectors on an {{mvar|n}}-dimensional vector space, is {{mvar|n}}&nbsp;choose&nbsp;{{mvar|k}}: {{nowrap|<math display="inline"> |\mathcal{J}_{k,n}| = \binom{n}{k}</math>.}} This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.