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I'm comparing two versions of Stokes' theorem, one for simplicial chains (as in Chapter 10 of Rudin's Principles of Mathematical Analysis) and the other for compact differentiable manifolds with boundary. Rudin notes that the chain version of Stokes' theorem can be applied to certain subsets of euclidean space that can be written as finite almost disjoint unions of simplexes. Namely, it applies to any subset $\Omega$ of $\mathbf{R}^{n}$ such that $$ \Omega = E_{1}\cup E_{2}\cup\dotsb\cup E_{r} $$ where each $E_{i}$ is the image of a one-to-one differentiable mapping $T_{i}$ of class $C^{2}$ in the standard $k$-simplex $Q^{k}$ and whose Jacobian is positive, and where the interiors of the subscripted $E$'s are pairwise disjoint. Such sets can be assigned a boundary chain, to which the chain version of Stokes' theorem can be applied.

I know that the any abstract differentiable manifold lies in a euclidan space. Is it possible to write any compact differentiable manifold with boundary in this manner?

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