heres a little example of stuff that came up and surprized me by its usefulness. I knew little about gauss and stokes, etc, but was interested inm topology and our department was having a seminar to learn de rham cohomology, sheaf version. then i had to teach several variable calculus.
somehow i began to wonder how one would prove thata circle really does wrap around the oprigin at the same time, and i noticed that stokes theorem i.e. greens thm was the right tool.
i.e. the fact that dtheta has integral 2pi around the circle implies the circle does not bound any disc that misses the origin, by greens thm.
i was so excited, i relaized this was actually the key idea behind de rham cohomology, but the people lecturing on the sheaf theory version of DR apparently did not know this.
i went on to include a proof that a sphere has no never zero vector fields by gauss thm in my calc class. later i saw an article on this topic in the American math Monthly, but not as elementary as the version I had discovered myself.
In my thesis i was studying the degree of a mapping of moduli spaces, and the usual technique for that is to use "regular values", but i did not have any at my disposal. It turned out the inverse and implicit function theorems could be applied to the normal bundle of a fiber to substitute for them.
This method was very effective, and had not been used before. It only came to mind because years before i had thought long and hard about those theorems from advanced calculus. I was using them in algebraic geometry but the ideas were the same, once understood deeply.
a beautiful technique in studying theta divisors of abelian varieties is to use the gauss map, as introduced by andreotti, or rather as re - interpreted by griffiths, from andreotti's proof. this is a classical idea used by gauss to measure curvature of surfaces in 3 space, but adapted by andreotti to study the geometry of jacobian varieties of algebraic curves.
fundamentally, the gauss map is an invariant of an embedded hypersurface (or more general surface). once understood, it becomes of interest to calculate it also for a divisor embedded in a complex torus, because like affine space, a complex torus has a trivial tangent bundle!
once you get past the nuts and bolts details of the gauss map and curvature in 3 space, and realize the gauss map is an invariant of all hypersurfaces in manifolds with trivial tangent bundle, you can use it more widely.
even in manifolds with non trivial tangent bundle, the "gauss map" taking a map on points, to its derivative, a map on tangent spaces, is of interest in measuring properties of maps, as developed again by griffiths and carlson as the method of infinitesimal variation of hodge structures, in many settings.
the beautiful and powerful techniques in analytic number theory show that zeroes and poles of complex holomorphic map[pings are intimatel connected to number theoretic proe\perties. see the proof by dircihlet of the reslt on primes in arithmetic progression, which also uses crucially group theory.
the idea behind groups is just that of symmetry, which is why it is useful in many places especially physics.
etc etc...any ideas you understand can be useful. so specialize, but try to understand as deeply as possible those ideas you encounter.
(And the chain rule is even more obvious in differential geometry!)