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.
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:
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.