what is the most useful topic of higher mathematics (beyond vector

What about computational courses? PDEs is more useful than most people think. Sure, you can look 'em up and solve them numerically, but being able to look at them determine behavior is useful. I think the key is probably just taking what interests you. More is better, generally speaking. Start studying for the math GRE subject test now. Comparably, it isn't bad, but speed is important.

Differential forms, calculus of manifolds.

>>8576795
Functional analysis

differential equations in general

>>8576795
Also if you're doing mech engineering, I'd suggest spending your time learning measure and integration theory, and various proof techniques.

t. Applied Math undergrad

>>8576795
>in terms of application

That largely depends on what you want to apply it to.

For mech engineering, I'd say numerical analysis or PDEs

>>8576829
is that basically behind the scenes calculus of variations?
>as real analysis is to calculus
>>8576815
Are these subsets of topology? why do you recommend this?
>>8576843
Fluid mechanics
>>8576832
thank you, are those subjects in real analysis?
I have already studied proof

OP here
I should mention that I want to do applications of dynamical systems to geophysical fluid mechanics and wave phenomena.

This is of course subject to change, as I am only an undergrad

While we are on the subject, what would be the best book on calculus of variations? Preferably with exercises and examples.

>>8576918
idk. never done it yet. try this:
www math umn edu/~olver/ln_/cv.pdf

>>8576852
Differentials is essentially the highest level of useful analysis in APPLIED math, notably physics.

It uses topology to generalize complexities in analysis that otherwise are mathematically verbose, and needlessly intricate.

Buy it also allows for greater depth in what you are describing. If calculus is the tool for undergrads to understand basic EMT, then differentials is the bigger, badder tool that actual research physicists use to describe their problems

>>8576811
For electrical engineering undergrad planning on grad school with focus on controls
>Fourier Series
>PDE
>Complex/Real analysis
>Vector/Tensor analysis
>Advanced Calculus/Algebra
Any other recommended subjects to study for that specific area of electrical engineering application?

>>8576995
I dig it. Thanks a bunch, anon.