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I heard about manifolds with boundaries, but I never heard about manifolds with boundaries and vertices except perhaps in Spivak's book. Take a solid cube. It's a 3-dimensional manifold with a boundary and 8 vertices. So I think manifolds with boundaries and vertices are natural objects of mathematics. Particularly I'd like to see a proof of Stokes' theorem on these manifolds. Are there books which treat these manifolds?

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See this post on… – Brett Frankel Apr 19 '12 at 4:11
@Brett Thanks a lot. – Makoto Kato Apr 20 '12 at 1:14

Papers, books and articles which treat manifolds with corners and Stokes' theorem on them:

Joyce "On manifolds with corners"

Partial Differential Equations 1. Foundations and Integral Representations by Friedrich Sauvigny.

John Lee's book "Introduction to smooth manifolds".

Brian Conrad's notes on differential geometry:

Ch. XXIII in Lang's Real and Functional Analysis entitled "Stokes' Theorem with Singularities".

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