Stokes' theoremtheorem
Formulated in the mid-19th century by George Stokes for hydrodynamics, the famous formula quickly became a universal tool of physics. The classical form: the circulation of a vector around a closed loop equals the flux of its curl through any surface spanning the loop: ∮_C F·dr = ∬_S (∇×F)·dS. The generalized version—the grand Stokes' theorem—states: the integral of a differential form over the boundary of a manifold equals the integral of its exterior derivative over the whole manifold. This formula elegantly unifies special cases: Green's theorem in the plane, Gauss–Ostrogradsky theorem in space, and the Kelvin–Stokes theorem itself.
How it works
The lift of a wing is a direct consequence of air circulation around the airfoil; Stokes' theorem relates it to the vortices shedding from the trailing edge. In an induction stove, an alternating magnetic field generates a vortex electric field in metal cookware, and currents heat it up—this is Faraday's law in integral form, equivalent to Stokes' theorem.
💡 The most concise expression of the fundamental theorem of calculus, ∫_a^b f'(x)dx = f(b)–f(a), is also a special case of the general Stokes' theorem, just for a one-dimensional interval.
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
∮_∂S — integral over the closed contour ∂S; F — vector field; dr — path element; ∬_S — integral over surface S; ∇×F — curl of F; dS — area element directed along the normal
\int_{M} d\omega = \int_{\partial M} \omega
∫_M — integral over manifold M; dω — exterior derivative of the k-form ω; ∫_∂M — integral over the boundary ∂M. The right side is interpreted as ‘flux through the boundary’