This chapter introduces just enough differential topology to serve as a suitable framework for Stokes's Theorem. The subject matter is the elementary structure of smooth manifolds, which is a topic in real analysis that sits at the intersection of algebraic topology and differential geometry.

Section 1 presents the beginning definitions and results about smooth manifolds, tangent vectors and vector fields, cotangent vectors and differential 1 forms, derivatives of smooth mappings, and differentials.

Section 2 defines the exterior algebra of a finite dimensional real vector space. Tensor algebras, which are discussed in Chapter VI of the author's Basic Algebra, are taken as known.

Section 3 introduces differential forms and their pullbacks under smooth maps. It shows how to compute pullbacks, and it establishes some properties of them.

Section 4 introduces the exterior derivative, which is the differentiation operator to be used with differential forms, and shows that it satisfies a number of properties.

Section 5 contains the construction of a smooth partition of unity, which is a device making it unnecessary in many cases to cut manifolds into pieces when treating integration problems.

Section 6 introduces the notion of an oriented smooth manifold and integration of top-degree differential forms on it. The section shows also the relationship between integration and pullback.

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.

This chapter looks for a single setting in which Stokes's Theorem applies at once to all situations of practical interest. It begins by developing the theory in the setting of manifolds-with-corners and continues with a theory in a more general setting studied by H. Whitney.

Section 1 introduces the model space $\bQ^m$ for $m\geq2$, in terms of which manifolds-with-corners are defined. The section contains one result that is relatively hard to prove: the index of a point of $\bQ^m$ is taken to be the number of coordinates that are equal to 0, and it is shown that any diffeomorphism between open sets in $\bQ^m$ maps points of one index into points of the same index. Consequently the notion of index is well defined for the points of a manifold-with-corners. Other definitions concerning manifolds translate easily into corresponding definitions for manifolds-with-corners. These include smooth real-valued function, support, germ, tangent space, cotangent space, smooth differential forms, pullbacks of differential forms, and the derivative of a smooth map between manifolds-with-corners.

Section 2 introduces strata, the stratum $S_k(M)$ consisting of all points of index $k$ in a manifold-with-corners $M$. Strata have a number of useful properties, one of which is that the strata of index 0 and 1 combine to yield a manifold-with-boundary.

Section 3 gives a version of the Stokes's Theorem for manifolds-with-corners, saying $\int_{\pa M}\gw=\int_Md\gw$ as usual. In this equality the integral on the left is over the stratum of all points of index 1, and the integral on the right is over the stratum of all points of index 0. Simple examples show that this theorem is not a trivial consequence of the theorem about manifolds-with-boundary when applied to the manifold-with-boundary consisting of all points of index 0 and 1 in $M$.

Section 4 establishes a version of the Divergence Theorem due to Whitney that applies to any bounded region of $\bR^m$ for $m\geq2$ when most of the topological boundary behaves as it does for a manifold-with-boundary and when the set of exceptional points of the topological boundary is small in a specific technical sense. Such a region will be called a Whitney domain. If the set of exceptional points is finite, then it is small in the technical sense.

Section 5 examines in some detail the technical condition in Section 4. That condition becomes: the set of exceptional points is compact and either is empty or has $m-1$ dimensional Minkowski content 0. It is shown that the condition that a compact set has $\ell$ dimensional Minkowski content 0 is intrinsic to the set as a subset of a Euclidean space and does not depend on its embedding. Furthermore any function from one Euclidean space to another that satisfies a Lipschitz condition always carries compact subsets of $\ell$ dimensional Minkowski content 0 to compact sets of $\ell$ dimensional Minkowski content 0. Consequently the notion "$\ell$ dimensional Minkowski content 0" is well defined for compact subsets of smooth manifolds and is preserved under smooth mappings into Euclidean spaces. The section concludes with examples of Whitney domains constructed from the zero loci of polynomials.

Section 6 extends the scope of Stokes's Theorem to Whitney manifolds, a class of spaces that includes all manifolds-with-corners and that allows all Whitney domains as additional model cases. The result is that the Stokes formula applies in what seems to be the full set of practical situations of interest to mathematicians, physicists, and engineers.