# Application of Stokes Theorem

1. Dec 2, 2009

### filter54321

1. The problem statement, all variables and given/known data
Use Stokes's Theorem to evaluate $$\int F · dr$$
In this case, C (the curve) is oriented counterclockwise as viewed from above.

2. Relevant equations
F(x,y,z) = xyzi + yj + zk, x2 + y2 ≤ a2
S: the first-octant portion of z = x2 over x2 + y2 = a2

3. The attempt at a solution
"Use Stoke's" is code for "stick the dot product of curl F and the normal vector into the integral". If this problem behaves nicely, this should become a double integral like every other Stoke's problem in the book and will need a new area of integration as well.

curl F would be <0, xy-yz, -xz>
G(x,y) = z-x2
the normal is <-Gx,-Gy,1>
which is <-2x, 0, 1>
F · N = 0+0-xz = -xz because of convenient canceling

So now -xz has to be integrated over some area. The "first octant" is simple enough, and x2 + y2 ≤ a2 is a cylinder of infinite height and radius a centered around the Z-axis (a fixed circle for every Z). I'm picturing a a parabola-cum-trough looking thing that got stamped by a circular cookie cutter.

If that's the case, the x's get restricted which means the z's get restricted to a constant. How do you set up the integral to get rid of all variables (the radius a is a constant and obviously stays undefined).

Thanks!