Multivariable calculus learning recommendations

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.