An Introduction to Multivariable Analysis, from Vector to Manifold, by Piotr Mikusiński and myself was published in 2002.  Though self-contained, we also thought of it as a sequel to An Introduction to Analysis: From Number to Integral by Jan and Piotr Mikusiński.    (Jan was Piotr’s father and a mathematician with a world reputation.)  The book by the Mikusińskis is an introduction to real analysis for functions of a single variable.

The idea of our book was to provide a fairly rigorous introduction to real analysis in the setting of .  Two features that made it different from other texts covering the same ground were, on the one hand, development of Lebesgue integration in a way that involved infinite series rather than measure theory and, on the other, a particularly geometric development of the wedge product.  Here are some reviews:

This is a self-contained textbook devoted to multivariable analysis based on nonstandard geometrical methods. The book can be used either as a supplement to a course on single variable analysis or as a semester-long course introducing students to manifolds and differential forms. Mathematical Reviews.

The authors strongly motivate the abstract notions by a lot of intuitive examples and pictures. The exercises at the end of each section range from computational to theoretical. The book is highly recommended for undergraduate or graduate courses in multivariable analysis for students in mathematics, physics, engineering, and economics. Studia Universitatis Babes-Bolyai, Series Mathematica.

All this [the description on the book’s back cover] is absolutely true, but omits any statement attesting to the high quality of the writing and the high level of mathematical scholarship. So, go and order a copy of this attractively produced, and nicely composed, scholarly tome. If you’re not teaching this sort of mathematics, this book will inspire you to do so. MAA Reviews.