Stokes Theorem
Statement. For an oriented $n$-manifold $M$ with boundary and $\omega \in \Omega^{n-1}(M)$ with compact support: $$\int_M d\omega = \int_{\partial M} \omega.$$
Intuition. One theorem containing the fundamental theorem of calculus, Green's theorem, classical Stokes, and the divergence theorem — the reason $d$ and the boundary operator $\partial$ are "adjoint."
Physics anchor. Flux–circulation relations of electromagnetism. Most important instance for this study: $$\oint_{\partial S} A = \int_S F,$$ holonomy of the potential around a loop = flux of the field through a spanning surface — the geometric core of the Aharonov-Bohm Effect (when no spanning surface stays in the field-free region, the loop integral survives even with $F = 0$ locally).
Related: Differential Form · Exterior Derivative · de Rham Cohomology · Parallel Transport and Holonomy · Module 02 - Differential Forms Refresher
Sources: Sternberg Ch. III.