Is Vector Calculus useful for pure math?

Question

I have the option to take a vector calculus class at my uni but I have received conflicting opinions from various professors about this class's use in pure math (my major emphasis). I was wondering what others thought about the issue. I appreciate any advice.

Answer 1

The answer depends on your interests, and on the place you continue your education. In some areas in the world, PhD's are very specialized so any course that is not directly related to the subject matter is not necessary. One could complete a pure math PhD and not know vector or multivariate calculus.

However, this is increasingly rare; more and more PhD programs are following the American model, expecting their graduates to have some breadth of knowledge in addition to the specialized skills required for the PhD thesis itself. Such programs would consider a lack of vector calculus a serious deficiency and may not even consider your application into any math PhD program, pure or applied. Certainly this is the case at most universities in the U.S.

Answer 2

Yes. "Studying math is as much about specific examples as it is about general theorems." My teacher Mike Artin used to expound on this concept (and then assign ridiculous and tedious homework assignments) and I would not get what he was saying. But the more I read math, the more I've come to realize that understanding specific examples is what allows you appreciate the power and beauty of general results.

Basic vector calculus, while it might not be immediately applicable to the next few classes you're taking, will provide you with a plethora of examples that will enrich your understanding of other concepts you encounter and enhance your "mathematical maturity". I also used to get confused when my teachers would talk about mathematical maturity, but a large part of it is simply having a large enough pool of examples and concepts to draw from so that you can fully appreciate certain results.

There are deep and surprising connections across many different areas of math. Some of these ideas will require you to understand how do calculus on manifolds. For example algebra and topology are considered two main areas of "pure math". So when you get around to studying algebraic topology, you might come across something like the de Rham cohomology, and then you'll think to yourself "Good thing I studied vector calculus back in the day."

Answer 3

My two cents: if you are not sure about where exactly in mathematics you want to go, then vector calculus is a pretty good way to make more paths viable. If you do any sort of analysis, chances are you'll need to know vector calculus forwards and backwards; this is especially true if you want to work in PDEs. If you want to do anything on manifolds (differential geometry, topology, etc.) you will need an excellent understanding of multivariate calculus.

Vector calculus gives you many mathematical objects you can play with - vector fields, line and surface integration, surface parametrizations, implicit and inverse function theorems, optimization techniques, etc. - that are fundamental objects of study elsewhere. Vector fields will pop up in differential equations, topology, and geometry; optimization in finitely many variables will give you valuable insight when you start thinking about optimization and analysis in infinite-dimensional spaces; the various forms of vector field integration will be vital to your understanding of differential equations; the inverse and implicit function theorems are very foundational theorems without which much of differential geometry and topology would be impossible; and vector calculus in 2 dimensions gives you a valuable example for comparison in complex analysis. This is not an exhaustive list, and even if you don't think you'll be studying anything related to any of the aforementioned topics (unlikely), understanding calculus in $\mathbb{R}^n$ gives you many examples to test your intuition and understanding throughout mathematics.

I don't know where you are in your mathematics education, but if you're trying to choose between vector calculus and something more focused in a specific direction of pure mathematics, I would suggest you choose vector calculus. It's more likely that you'll use ideas from vector calculus in other areas of mathematics than vice versa; e.g. I can imagine applying ideas of vector calculus in number theory (through complex analysis) more easily than applying number theory to vector calculus.