The author, Clark Bray, is an Associate Professor of the Practice in the Mathematics Department at Duke University. These positions are prized appointments among university faculty positions that are both (1) at top research schools and (2) primarily focused on teaching instead of research. In this role, Prof. Bray has been the course supervisor for multivariable calculus (Math 212) at Duke for several years.

More about Prof. BrayOn Prof. Bray's YouTube channel (Clark Bray Math), one of the playlists includes a complete set of lectures from his course in multivariable calculus at Duke University. These lectures are taught directly from this textbook, and will help you understand all of the ideas in the book. Subscribe now to start learning!

See the YouTube lecturesAlso included in the playlist on Prof. Bray's YouTube channel are hundreds of videos with explanations of the book exercises. These were recorded as Prof. Bray was explaining the solutions to actual Duke students in his course, and give thorough discussions of the context of each question and related ideas.

See the YouTube exercisesTextbook prices are soaring these days. This textbook, by publishing directly through Amazon with no traditional publishing company to inflate prices, is available for a fraction of the typical cost. Instructors, you can collectively save your students thousands of dollars by using this book. And for students taking a course with a different textbook, you can supplement your learning resources dramatically with this book, and all of the YouTube lectures and exercises, for probably only a small fraction of what you paid for your existing book.

Order on AmazonMany math books show definitions, theorems, and equations, but don't put very many words into explaining the objects and ideas that are being used. This book gives long discussions, to help students get a sense for what is happening and develop intuition for the ideas being discussed. This is part of the reason that the book is so long, at over 480 pages!

This text focuses on giving students intuition for the geometry of the ideas and for the strategies for solving problems. There is less focus on rigor, which for most students at this level is not as important.

Students using this book can benefit even more by looking through actual Duke exams, with solutions, from Prof. Bray's class using this textbook. Use the blank exams as practice material and look at the solutions as you go; or, use the blank exams to give yourself a mock testing situation, and then compare your work with the solutions to gauge your progress.

See the exam materialsMost math books have figures that are created by computer graphics programs. The results look good, but they give no help to students who are trying to learn to make their own figures while working in the subject. This book's figures are drawn by hand, giving students a reasonable target to try to reproduce and learn from.

Each section ends with a long list of exercises to give students practice to help solidify what they have learned. The exercises are of a range of difficulties, from easy computations using given formulas to creative applications requiring insight and clever uses of the tools. If you get stuck, just check the YouTube channel to see solutions!

See the YouTube exercises