next up previous contents
Next: Conservative Fields, Potential Function. Up: Integration and Vector Fields. Previous: The Divergence theorem.

Stoke's theorem.

For our surfaces, we will use parametrizations (v.s. section  2).

Theorem 8.38   Let $\overrightarrow{F} =M \overrightarrow{i} + N \overrightarrow{j} + P \overrightarrow{k} $ be a vector field defined over a domain in space.

Let $\mathcal{S}$ be a surface whose boundary is the curve $\mathcal{C}$ in space; we denote $\overrightarrow{n} $ a unit vector, normal to the surface $\mathcal{S}$.


\begin{displaymath}\oint_{\mathcal{C}} \overrightarrow{F}\cdot d \overrightarrow...
...\times \overrightarrow{F}\cdot \overrightarrow{n}\; d \sigma .

Example 8.39  

Example 8.40   Let $\overrightarrow{F} =(z-y,x-z,y-x)$. We wish to compute the work of $\overrightarrow{F} $ along the intersection $\mathcal{C}$ of the cylinder whose equation is x2+y2=1 with the plane whose equation is x+y+2z=4.

Denote $\mathcal{S}$ the interior of the ellipse $\mathcal{C}$. We have:

Noah Dana-Picard