Update 02/07/2017:

The original book's URL at Kenneth Kutler's homepage in the Department of Mathematics at Brigham Young University is no longer accessible. There are other copies lying around the internet, but I couldn't be sure about the legality of these copies. Thus, I decided not link to them and remove this book entry instead.

Update 10/08/2017:

The book has been made publicly accessible again at Kenneth Kutler's homepage in the Department of Mathematics at Brigham Young University. The download link has been updated.

Excerpts from the Introduction:

Multivariable calculus is just calculus which involves more than one variable. To do it properly, you have to use some linear algebra. Otherwise it is impossible to understand. This book presents the necessary linear algebra and then uses it as a framework upon which to build multivariable calculus. This is not the usual approach in beginning courses but it is the correct approach, leaving open the possibility that at least some students will learn and understand the topics presented. For example, the derivative of a function of many variables is a linear transformation. If you don't know what a linear transformation is, then you can't understand the derivative because that is what it is and nothing else can be correctly substituted for it. The chain rule is best understood in terms of products of matrices which represent the various derivatives. The concepts involving multiple integrals involve determinants. The understandable version of the second derivative test uses eigenvalues, etc.

The purpose of this book is to present this subject in a way which can be understood by a motivated student. Because of the inherent difficulty, any treatment which is easy for the majority of students will not yield a correct understanding. However, the attempt is being made to make it as easy as possible.

Many applications are presented. Some of these are very difficult but worthwhile. Hard sections are starred in the table of contents. Most of these sections are enrichment material and can be omitted if one desires nothing more than what is usually done in a standard calculus class. Stunningly difficult sections having substantial mathematical content are also decorated with a picture of a battle between a dragon slayer and a dragon, the outcome of the contest uncertain. These sections are for fearless students who want to understand the subject more than they want to preserve their egos. Sometimes the dragon wins.