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In this chapter we begin our discussion of integration on manifolds. We introduce the notion of a singular cube in a manifold and explain how a differential form can be integrated over such a cube. We then mirror the construction of singular homology and show how to create the (smooth) cubical chain complex of M. We will use this formalism next chapter en route to proving the manifold version of Stokes’ Theorem.

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Smooth Singular Cubes

  • 章节

  • 发表日期：

    2026年

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  • Will J. Merry

摘要

In this chapter we begin our discussion of integration on manifolds. We introduce the notion of a singular cube in a manifold and explain how a differential form can be integrated over such a cube. We then mirror the construction of singular homology and show how to create the (smooth) cubical chain complex of M. We will use this formalism next chapter en route to proving the manifold version of Stokes’ Theorem.