Stokes' Theorem

Definition

Suppose we have a smooth surface $S$ in three-dimensional space ${\mathbb{R}}^3$, which can be parameterized by some function $\vec{S}(u,v)$, and that this surface’s boundary is represented by a curve, $C=\partial S$, which can be parameterized by the function $\vec{C}(t)$. Note that the statement $C=\partial S$ defines $C$ as the infinitesimal new area added to $S$ if $S$ were to grow outwards along its border by some infinitesimal amount. Suppose we also have a vector field $\vec{F}(x,y,z)=⟨F_x(x,y,z),F_y(x,y,z),F_z(x,y,z)⟩$ permeating this space ${\mathbb{R}}^3$ so that all of $S$ is within it, and that $\vec{F}$ has continuous first-order partial derivatives on $S$. Then Stokes’ Theorem states,

$${\iint }_S(\nabla \times \vec{F})\cdot d\vec{S}={\oint }_{\partial S}\vec{F}\cdot d\vec{C}$$

Derivation

Stokes’ Theorem unifies two methods of calculating circulation around a surface. Suppose we have a surface $S$ existing in the presence of a vector field $\vec{F}$: if we wanted to calculate

References

1. Unit 23: Stokes’ Theorem - Harvard University
2. A Derivation of Stokes’ Theorem by Zaché, Luchini, and Battisti