# Math Help - Stoke's Theorem

1. ## Stoke's Theorem

Use the surface integral in Stoke's Theorem to calculate the circulation of the field F around the curve C in the indicated direction: F=x^2y^3i + j + zk, C: The intersection of the cylinder x^2+y^2=4 and the hemisphere x^2+y^2+z^2=16 , z greater or equal to 0, counter clockwise when viewed from above.

first we see that $\text{curl}(\bold{F}) = -3x^2y^2 \bold{k}.$ we have $S: \ z=2\sqrt{3}$ and $D=\{(x,y): \ x^2+y^2 \leq 4 \}.$ clealy, the upward normal unit of $S$ is $\bold{n}=\bold{k}.$ also we have $dS=dxdy.$ therefore:
$\int \int_D \text{curl}(\bold{F}) \cdot \bold{n} \ dS=-3 \int \int_{x^2 +y^2 \leq 4} x^2y^2 \ dxdy=-8 \pi.$