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As part of an engineering degree, I've taken courses on single variable calculus, linear algebra, multivariable/vector calculus and ordinary differential equations. I'm also about to study an advanced engineering mathematics course, which covers three topics (albeit quite superficially): Complex variable calculus, fourier series/transform and partial differential equations.

Seeing how multivariable/vector calulus generalizes concepts like limits, derivatives and integrals in elegant and beautiful ways, and introduces new concepts like vector fields that feel natural and intuitive, I was wondering if there's a branch in mathematical analysis that does the same thing, extending multivariable calculus to something else, which I would be able to study by myself using the knowledge I have from the courses previously mentioned.

I've heard about things like tensor calculus, calculus of variations and differential geometry, but I'm not sure if any of those would be what I'm looking for (a logical next step form multivariable calculus). Any recommendations on books for self-studying the subject/subjects you consider appropiate are greatly appreciated.

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    $\begingroup$ Don't forget to take some probability course. $\endgroup$ – mathreadler Sep 18 '17 at 20:06
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    $\begingroup$ You don't mention what kind of an engineer you would want to become ;) For an EE signal processing engineer, the next step would probably be Fourier analysis or transform theory. Transforms are used literally everywhere these days and will stress your linear algebra and calculus understanding simultaneously. $\endgroup$ – mathreadler Sep 18 '17 at 20:13
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    $\begingroup$ I would also strongly recommend some numerical analysis, especially numerical linear algebra. $\endgroup$ – icurays1 Sep 18 '17 at 20:14
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    $\begingroup$ Also some introductory optimization course if you haven't had any so far. $\endgroup$ – mathreadler Sep 18 '17 at 20:24
  • $\begingroup$ Forgot to mention, I've also taken a probability course and will be taking a numerical analisys course a couple of semesters from now. Thanks for the heads up. Mechatronics engineering, btw. $\endgroup$ – JuanEsteban Valdez Sep 18 '17 at 21:28
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The area of mathematics that can be considered as a (the) natural extension or generalisation of multivariate calculus is differential geometry. You can see this extension carried out, while staying in the context of Euclidean spaces, in for instance:

  1. Dineen
  2. McInerney

More generally, a very nice introduction to differential geometry us given by Tu.

It sounds like you would have the prerequisites to work through any of these books.

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    $\begingroup$ These are all very sophisticated, requiring some multivariable analysis and topology. You might check out my free differential geometry text, linked in my profile, which treats just curves and surfaces in $\Bbb R^3$, written for students who've had the background you have. It's far more concrete and hands-on. $\endgroup$ – Ted Shifrin Sep 18 '17 at 21:21

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