Stoke’s Theorem

• A higher dimensional version of Green’s Theorem.
• Allows you to calculate a line integral using a surface integral (flux)

Formula

$${\oint }_{\partial S}\vec{F}\cdot d\vec{r}={\iint }_S\vec{\nabla }\times \vec{F}\cdot \hat{n}dS$$

$\vec{n}$ needs to be positively oriented

Positive Orientation

• If you draw a counter clockwise circle around a normal vector and it is able to be deformed to make the direction of $C$
• While you are walking along $C$ if your head is pointing in the same direction as the unit normal vectors while the surface is on the left

Surface as Graph

$z=g(x,y)$

$${\iint }_S\vec{\nabla }\times \vec{F}\cdot \hat{n}dS={\iint }_D{\vec{\nabla }\times \vec{F}\cdot \vec{n}|}_{z=g(x,y)}dA$$

Changing Contours in 3D

1. Pick any $S$ such that $\partial S=C\cup (-C^{\prime })$
2. Choose $\hat{n}$ accordingly
3. ${\oint }_{\partial S}\vec{F}\cdot d\vec{r}={\iint }_S(\vec{\nabla }\times \vec{F})\cdot \hat{n}dS$

$${\int }_C\vec{F}\cdot d\vec{r}={\int }_{C^{\prime }}\vec{F}\cdot d\vec{r}+{\iint }_S(\vec{\nabla }\times \vec{F})\cdot \hat{n}dS$$