Chapter II. Manifolds-with-Boundary

Abstract

This chapter introduces oriented manifolds-with-boundary, obtains Stokes's Theorem for them, and shows that the classical theorems of Green, Gauss-Ostrogradsky, and Kelvin–Stokes fit into this framework.

Section 1 introduces the subject by working with ordinary oriented smooth manifolds, i.e., those oriented smooth manifolds without boundary. Stokes's Theorem for this situation reduces to a theorem about compactly supported differential forms in Euclidean space.

Section 2 introduces smooth manifolds-with-boundary of dimension $m$, charts being homeomorphisms from nonempty open subsets of the manifold-with-boundary onto relatively open subsets of the closed half space $\bH^m$ of $\bR^m$. One distinguishes manifold points and boundary points and observes that the set of manifold points yields a smooth manifold. The section defines smoothness of real-valued functions and associated objects, and for this setting, it goes through much of the same kind of development that was done for manifolds in Chapter I.

Section 3 defines orientability of a smooth manifold-with-boundary to mean orientability of the smooth manifold of manifold points. If a smooth manifold-with-boundary is orientable, then so is its boundary, and a particular choice of orientation of the boundary, known as the induced orientation, is defined so that the signs will eventually work out properly in Stokes's Theorem.

Section 4 states and proves Stokes's Theorem for oriented smooth manifolds-with-boundary, handling the case of dimension $m=1$ separately from the case of dimension $m\geq2$.

Section 5 examines the meaning of Stokes's Theorem in the settings that give rise to three classical integration theorems—Green's Theorem, the Divergence Theorem, and the Kelvin–Stokes Theorem—and in the setting of line integrals independent of the path.