Stokes' theorem: add up the swirl across a surface to get the circulation at its edge
Picture a vector field crossing a surface bounded by a closed loop. Stokes' theorem says the total circulation around that loop equals the sum of all the local rotation, or curl, across the surface inside it. A calculation along an edge becomes one over an area, and the reverse works too.
Also known as the Kelvin–Stokes theorem, the curl theorem, the rotor theorem or the fundamental theorem for curls, it is a result of vector calculus. One side of the equation is a line integral of the field taken around the boundary curve; the other is a surface integral of the field's curl over an oriented surface enclosed by that curve. In its usual form the field needs continuous first-order partial derivatives.
An elementary proof boils the three-dimensional statement down to Green's theorem, its two-dimensional counterpart, in four steps, starting with a parametrisation of the surface by a flat region and a pullback of the field onto it. Mathematicians more often derive it as a special case of the generalized Stokes theorem, stated with differential forms, where the vector field becomes a 1-form and its curl becomes the exterior derivative of that form.
The subtle part is saying exactly what a boundary is. A shape like the Koch snowflake has no boundary that can be integrated in the Riemann sense, and Lebesgue theory cannot assign a surface measure to a surface that is not Lipschitz. One advanced route uses a weak formulation handled with geometric measure theory and the coarea formula. A simpler route starts in the plane, where the Jordan curve theorem guarantees that a simple closed curve splits the plane into a compact piece and a non-compact piece, then carries that picture onto the surface through its parametrisation.
The theorem also sits inside a broader identity involving an extra field g, and the ordinary version drops out when g is a uniform scalar field.
Source: Stokes' theorem
