Is Vector Calculus useful for pure math?

[jameseg]

Hi everyone, 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.

 

[micromass]

It most certainly is useful. For example, in differential geometry.

That said, there might be classes out there which teach the material better. For example, an analysis on manifolds class would probably be better suited for you.

 

[Tac-Tics]

Having an intuition about linear algebra will help enormously if you take abstract algebra. Vectorspaces are just modules over the real (or the complex) numbers. You can immediately relate group and ring homomorphisms to linear maps. Kernels are just the nullspace, the dimension is closely related to the notion of generators, most algebraic structures have the same notion of a direct sum.

Plus, linear algebra is often used as an easy introduction to elementary proofs.

 

[micromass]

Having an intuition about linear algebra will help enormously if you take abstract algebra. Vectorspaces are just modules over the real (or the complex) numbers. You can immediately relate group and ring homomorphisms to linear maps. Kernels are just the nullspace, the dimension is closely related to the notion of generators, most algebraic structures have the same notion of a direct sum.

Plus, linear algebra is often used as an easy introduction to elementary proofs.

Very true, but what does this have to do with vector calculus?

 

[mathman]

Hi everyone, 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.

Pure math covers a lot of ground. There are areas where vector calculus is important (various branches of analyis), while there are others (algebraic areas) where it may not be.

 

[ArcanaNoir]

Yes you should take it. You never know what you'll get into in grad school. For example I had always imagined I'd avoid analysis completely, until the school I was admitted to required an analysis qualifier. Good thing I forced myself through two semesters of analysis already. Yes vector calc might be very important, or you might not use it so much. But you should be prepared :)