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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 BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück