# What courses require multivariable analysis?

For which undergraduate and graduate mathematics courses is multivariable analysis* an essential prerequisite?

$\text{*}$ That is, a rigorous follow up to a first real analysis course at the level of baby Rudin.

By multivariate analysis, I'm assuming that you're talking about a course that covers the following topics in a rigorous way:

• Differentiation of functions between vector spaces

• The inverse and implicit function theorems, and related topics (e.g. rank of the derivative, immersions and submersions).

• Critical points and regular points of multivariable functions

• Multiple integrals and some version of Fubini's Theorem

• Differential forms and Stokes' Theorem

• Possibly an introduction to differentiable manifolds

My opinion is that such a course isn't an essential prerequisite for any later courses, for the simple reason that many (or perhaps most) colleges do not offer a course on multivariate analysis.

That being said, I think such a course would be very helpful for students who are planning to go into geometry, topology, or related subjects. A typical graduate course on differentiable manifolds (out of, say, Lee's Introduction to Smooth Manifolds) covers some of this material very quickly near the beginning, and covers differential forms quickly in the middle and in the more general setting of manifolds. As with many graduate courses, such a course is theoretically self-contained, but it helps a lot to have seen some of the material before. Other graduate courses that use this material might include courses on Riemannian Geometry, Differential Topology, Algebraic Topology (out of, say, Bott and Tu), Lie groups, or even courses on Riemann Surfaces or Algebraic Geometry.

An undergraduate differential geometry course might also assume some of this material, but probably at most on the level of multivariable calculus. For example, it might assume that students have seen the inverse function theorem, but it wouldn't assume that students have seen the proof. This is probably true of almost any undergraduate course.

• Is such kind of course called 'multivariate calculus' in the anglo-saxon world of mathematics? – Roland Apr 16 '14 at 9:26
• In the United States, "multivariable calculus" would refer to a course that covers partial derivatives and multiple integrals, as well as vector fields, divergence, curl, line and surface integrals, Green's theorem, the divergence theorem, and Stokes' theorem for surfaces in $\mathbb{R}^3$ (but probably not differential forms). The students would be a mix of math majors, science majors, economics majors, and possibly engineers, and they wouldn't write any proofs--the focus would be on computation. Such a course would typically be taken in the first or second year of college. – Jim Belk Apr 16 '14 at 14:39
• To follow-up on Jim Belk's comment, it used to be common (in the U.S.) to offer a 2-semester upper level course called advanced calculus, and part of that course dealt with a rigorous treatment (typically generalized to ${\mathbb R}^{n})$ of the lower level multivariable calculus material. This course has now largely been replaced by a single semester undergraduate real analysis course. Depending on the text used (e.g. Edwards' Advanced Calculus of Several Variables or Fleming's Functions of Several Variables), the multivariable part could be substantial or not very much. – Dave L Renfro Apr 16 '14 at 16:43
• "Advanced Calculus" has had so many different meanings across the various universities over the years that the term is effectively meaningless. – WetlabStudent Jun 15 '14 at 1:07

Motivated by Jim Belk's answer, let me mention that a rigorous advanced PDE course builds heavily on the material you mention. In particular,

• generalizations of integration by parts (Gauss/Green/Stokes..) are used for the weak formulation of PDEs.
• Technical theorems about Sobolev spaces and traces use elementary facts of basic differential geometry (partition of unity, local coordinates, etc.)
• Variants of the implicit function theorem are used in many local existence theorems (like the Nash-Moser iteration technique)

"Multivariable analysis" is the key to "advanced calculus." The latter includes (but is not limited to):

• Differential equations, especially if they include partial differential equations
• Vector and matrix calculus
• Real and complex analysis