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Definition: Differential Form of Degree k
Let \(M\) be a differentiable manifold. A differential form of degree \(k\) is a section of the \(n\)th alternating product of its cotangent bundle, i.e. it is a function
\[\omega \colon \cases{M\longrightarrow \bigwedge ^{k}T^{ * }M,\\x\longmapsto \omega (x),}\]
with \(\omega (x)\in \bigwedge ^{k}T_{x}^{ * }M\).
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References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck