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# Multivariable calculus learning recommendations

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This page provides subject-specific learning recommendations for the subject multivariable calculus. See all our learning recommendations pages |

Multivariable calculus is an important mathematical topic that studies functions of several variables from a calculus-based perspective (limits, continuity, differentiation, and integration). The course may be taken as follows:

• It is typically taught after single variable calculus in college. In colleges with semester systems, the course is often labeled Calculus 3, whereas in colleges with quarter systems, it may be labeled Calculus 4.
• Some students who intend to major in mathematics may not take a separate multivariable calculus course but may see it as part of an analysis course.
• Students who finish AP Calculus in 11th grade (junior year) may see it in 12th grade (senior year) in school (see our high school mathematics page for more details on the typical course sequences taken by high school students).

## Parts of multivariable calculus and their relative importance

Part of multivariable calculus Details Who needs to learn this?
Foundation Basic computational procedures and conceptual meaning of limits, partial derivatives, and integrals for functions of multiple variables, some basics of gradient vector and directional derivatives Anybody who needs to learn multivariable calculus for any purpose. This includes people who will see mathematical models in the natural sciences and social sciences.
Optimization Optimization using the Hessian test, Lagrange multipliers Most people who will see mathematical models in the natural and social sciences. It is of particular importance for people doing machine learning.
Linear algebra-based multivariable calculus The use of ideas from linear algebra to understand differentiation in multivariable calculus more deeply, including a more in-depth treatment of the gradient vector, directional derivatives, Hessian, Jacobian, etc. Of use to people in more physically or visually oriented subjects, such as physics.
Vector calculus Goes deeper into the subjects touched on in linear algebra-based multivariable calculus above. Also covers divergence, curl, Green's theorem and Stokes' theorem. Largely of use to people who will study mathematics or physics. Might be of some use in the other physical sciences as well, but not in the social sciences.

## Recommendations

### Recommendations for multivariable calculus that is theoretically or physically oriented

This includes a good treatment of vector calculus and the geometric intuition behind multivariable calculus. Although many multivariable calculus textbooks ostensibly cover vector calculus, their treatment of geometric topics is generally not too good relative to other subjects. It is also harder to convey visual ideas through a textbook. We therefore make video-centric recommendations below.

• The Center of Math multivariable calculus text and videos (textbook, videos): Although we don't have direct experience with the Center of Math multivariable calculus textbook, the videos seem decent and cover the topics well, and the synchronization between the videos and text might help students.
• The MIT Fall 2010 Multivariable Calculus course, which includes video lectures, might be useful. This also uses an ostensibly geometric approach, and the integration with other course materials available online might help students, though our rough impression is that the videos are not as good as the Center of Math videos.

We also recommend the book Div, Grad, Curl, and All That: An Informal Text on Vector Calculus as a supplement to the standard material covered in vector calculus courses.

### Recommendations for multivariable calculus for social science majors (focused on the foundation and optimization parts)

• Most multivariable calculus texts treat the foundation and optimization parts of the multivariable calculus text reasonably well, largely because it's harder to mess up these topics. An example is Stewart's multivariable calculus text (part of his full calculus text). If you already own or have access to Stewart's book, that should be suitable. However, there is nothing special or unique about Stewart's book, so a cheaper alternative would be fine if you have one.
• You might also wish to look at the lecture notes, quizzes, and related videos for Vipul Naik's Math 195 course. Lecture notes and quizzes cover all the foundation and optimization topics, and videos are available on some of the more important subtopics.