Stokes's Theorem and Whitney Manifolds is mathematically the third volume in a five-volume series that systematically develops concepts and tools in algebra and real analysis that are vital to any mathematician, whether pure or applied, aspiring or established. Four of the volumes were written earlier, namely the pair Basic Algebra and Advanced Algebra and the pair Basic Real Analysis and Advanced Real Analysis.
Stokes's Theorem and Whitney Manifolds is mathematically the third volume in a five-volume series that systematically develops concepts and tools in algebra and real analysis that are vital to any mathematician, whether pure or applied, aspiring or established. Four of the volumes were written earlier, namely the pair Basic Algebra and Advanced Algebra and the pair Basic Real Analysis and Advanced Real Analysis. These volumes treated algebra and real analysis separately. The present volume, which depends significantly on Basic Algebra and Basic Real Analysis, cuts across the two subjects and addresses some theorems about integration in several variables that go under the general name Stokes's Theorem and that generalize the Fundamental Theorem of Calculus in one variable. They all state the equality of the integral of one function over a region or surface with the integral of a related function over the boundary. Theorems of this kind are of particular importance to physicists and engineers because of their usefulness in fluid flow and electromagnetic theory.
In fact, three classical versions of Stokes's Theorem known as Green's Theorem in the plane, the Divergence Theorem or Gauss–Ostrogradsky Theorem, and the Curl Theorem or Kelvin–Stokes Theorem are of such importance that physicists and engineers often insist that mathematicians include these theorems as part of the undergraduate mathematics curriculum even though a genuine understanding of the theorems requires much more preparation than one might at first expect. For a rectangular region Green's Theorem and the Gauss–Ostrogradsky Theorem easily reduce to the one-dimensional case. The rectangular case and some specific kinds of other regions are all that one can usually manage at the undergraduate level. The resulting discussion leaves one without a grasp of what properties of the geometry are important.
Elie Cartan in the 1930s discovered that the formulas in the three theorems are all special cases of a formula that could be expressed easily in terms of differential geometry and differential topology, and he proved the general formula in arbitrary dimensions under the hypotheses that the boundary is smooth and that a condition of orientability is satisfied. The proof is easy once the relevant background from differential geometry and differential topology has been established. This general formula is what is called Stokes's Theorem in the title.
The setting for Cartan's work is that of a smooth "manifold-with-boundary," and unfortunately rectangular regions are not smooth manifolds-with-boundary. The first two chapters of this book develop the relevant background and establish Stokes's Theorem for smooth manifolds-with-boundary. They build on parts of the author's Basic Real Analysis and Basic Algebra.
For applications one wants the theorem to be valid in more settings than the smooth case and the rectangular case. The last chapter of the book arrives at a setting due to Hassler Whitney that seems to handle all cases of interest to most physicists, mathematicians, and engineers. Before introducing Whitney's setting, the book reproduces a number of easy variants of Cartan's result that are often invoked to handle some but not all cases of interest. Most notable among these is the setting of smooth "manifolds-with-corners," which includes both the smooth case and the rectangular case. Unfortunately even as nice an object as a solid square pyramid in three-dimensional space is not a manifold-with-corners. Finally the book establishes the Divergence Theorem in Whitney's setting, in which it is assumed that the boundary of the region is like that of a smooth manifold-with-boundary except on a set that is small is a specific sense. Whitney's general form of Stokes's Theorem follows readily. These matters are explained in a little more detail in the Introduction, which follows the Preface.
This book differs from the other four in the series in that it has not been through a normal external refereeing process. It simply amounts to additional chapters prepared for some existing books. No one besides the author checked the mathematics before publication, and no one besides the author gave an opinion of how important the mathematics is. The author invites corrections and other comments from readers. He plans to maintain a list of known corrections on his own Web page for as long as he is able.