This is intended as a text for a second course in real analysis at the senior or first-year graduate level, covering functions of several variables, and assuming the student has already completed a good course in functions of one real variable. The first half of the book is a rigorous study of differentiation and integration in Euclidean spaces, and is similar to the coverage in many advanced calculus books, such as Apostol’s Mathematical Analysis. There is the Inverse Function Theorem, the Implicit Function Theorem, and various kinds of multiple integrals and change-of-variable formulas. The second half of the book deals with differential forms and calculus on manifolds, working toward the general form of Stokes’s Theorem for n-dimensional space.
A limitation of the book is that it deals only with submanifolds of Euclidean spaces (except for an appendix that sketches the general case in metric spaces). I think this is a reasonable approach for this kind of course. The author makes the exposition easy to follow by gradually building up the types of manifolds, first dealing with parallelepipeds, then open sets, then parameterized manifolds, then general manifolds. The book also helpfully refers back often to the special cases of functions on the real line and the well-known vector operators (div, grad, curl) in 3-space.
The most conspicuous weakness of the book is the exercises, which are not very challenging. They are mostly to work specific examples or to tie up loose ends in the main exposition.
A book with somewhat similar coverage is Spivak’s similarly-titled Calculus on Manifolds, although that book goes to the other extreme and consists of a concise narrative with very difficult problems; the problems are supposed to be the focus of the book. Much of the present book’s material is also in Rudin’s Principles of Mathematical Analysis, although in Rudin’s characteristically concise form, and Rudin stops at chains and so does not work in the same generality of manifold as handled here.
Bottom line: a very good concrete introduction to manifolds, plus some more conventional material on multivariable calculus.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.