Stokes theorem, parametrizing composite curves

Gauss M.D.

Homework Statement

Calculate the line integral:

F = <xz, (xy2 + 2z), (xy + z)>

along the curve given by:

1) x = 0, y2 + z2 = 1, z > 0, y: -1 → 1
2) z = 0, x + y = 1, y: 1→0
3) z = 0, x-y = 1, y: 0 → -1

The Attempt at a Solution

I don't think the problem is very difficult when just dividing the line integral into three parts, calculating each separately. But I want to be thorough to see if I got all the concepts.

I tried to draw the curve (see attachment) which made me realize a cone cut in half would be a capping surface, so we should be able to apply Stokes theorem. But I'm having trouble parametrizing it since we've basically dealt exclusively with very standard parametrizations.

I think that one parameter should be the height of the cone, h = $\sqrt{y^{2}+z^{2}$ running from 0 to 1 and the other should be the angle in the xz-plane running from 0 to $\pi$. I'm just having trouble setting up the variable substitution. Can anyone give me a push?

Attachments

• semicone.jpg
2.3 KB · Views: 298

Homework Helper
Hi Gauss M.D.!

Before you spend time trying to apply stokes …

what is the curl?

Gauss M.D.
<x-2,x-y,y^2>... you're saying Stokes theorem is a bad idea here?