Stokes Theorem
Table of Contents
1. What is Stokes Theorem
Let \(\Sigma\) be a smooth oriented surface in \(\mathbb{R}^3\), parameterized by \(\Phi(u,v)\), with boundary \(\partial \Sigma\equiv \Lambda\), which is parameterized by \(\phi(t)\). If a vector field \(\mathbf{F}\) has continuous first-order partial derivatives in \(\Sigma\), then
\[ \iint_\Sigma (\nabla\times \mathbf{F})\cdot\, d\Phi=\oint_{\partial \Sigma} \mathbf{F}\cdot\, d\phi \]
1.1. Intuitive Understanding
We may understand this theorem as: the total “spinning” (circulation) around a closed boundary of surface equals the sum of all the tiny “spins” over the entire surface.
1.2. Proof
The surface \(\Sigma\) is parameterized as \(\Phi(u,v)=\Big( x(u,v), y(u,v), z(u,v) \Big)\), then the line integral