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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Papers, books and articles which treat manifolds with corners and Stokes' theorem on them:
Joyce "On manifolds with corners" arxiv.org/abs/0910.3518.
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: math.stanford.edu/~conrad/diffgeomPage/handouts.html
Ch. XXIII in Lang's Real and Functional Analysis entitled "Stokes' Theorem with Singularities".