# Stoke's Theorem

## Homework Statement

Use Stoke's Theorem to calculate $\int_C\vec{F}\cdot\, dr$, where
$\vec{F}=<x^2z\, ,xy^2\,,z^2>$ and C is the curve of the intersection of the plane $x+y+z=1$ and the cylinder
$x^2+^2=9$.
(C is oriented clockwise when viewed from above.)
Answer: $\frac{81}{2}\pi$.

Stoke's Theorem

## The Attempt at a Solution

Okay, let me try to explain where I am getting lost. Firstly, I know that the premise of Stoke's Theorem is that is relates a line integral to a Surface integral.

When I graph this, I get a cylinder that is symmetrical about the z-axis and it is intersected by a plane which results in an ellipse. (see terrible drawing below)

Now, I need to parametrize (how do you spell that anyway?) S. Now S is the surface that is bounded by C right? If not, please stop me here.

Homework Helper
Now, I need to parametrize (how do you spell that anyway?) S. Now S is the surface that is bounded by C right? If not, please stop me here.
Yes. S is any surface that is bounded by C. Usually the easiest surface to work with is the plane surface bounded by C if C can be visualised as lying on a plane.