Stokes theorem

$\int {\int }_{\Sigma }(\nabla \times F)\cdot d\Sigma ={\oint }_{\partial \Sigma }F\cdot d\Gamma$

The above can be made explicit as:

$\int {\int }_{\Sigma }\left(\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)dydz+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)dzdx+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)dxdy\right)={\oint }_{\partial \Sigma }\left(F_{x}dx+F_{y}dy+F_{z}dz\right)$, where

• $\Sigma$ is a smooth oriented surface
• $\Gamma :=\partial \Sigma$ is the boundary of $\Sigma$
• $F(x):{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a Vector field

The integral of a vector field’s divergence measures the flow through the surface under the given force, meaning that it matches the boundary integral of said force.

Proof

• Find a proof of Stokes Theorem