Stokes theorem help

1. May 7, 2006

dzza

Hi, i cant seem to figure out how stokes theorem works. I've run through a lot of examples but i still am not having any luck. Anyway, some advice on a particular problem would be greatly appreciated.

The problem is: F(x,y,z) = <2y,3z,-2x>. The surface is the part of the unit sphere in the first octant; the normal vector n is directed upward.

I get that the curl of F is <-3,2,-2>. What I tried next was writing the equation for the sphere as z = f(x,y) = sqrt(1-x^2-y^2) and finding from that fx(x,y) and fy(x,y). I then tried evaluating the double integral in cylindrical coordinates over R of (3fx-2fy-2)dA, where R is the region from 0 to pi/2 and r = 0 to r=1. I changed all the x's and y's to their polar equivalents and didn't forget the r in the dA or anything. I got nothing close to the right answer.

I understand that the way i approached it might be flawed, so if you can help in either helping me understand why what i was doing is wrong or if you have a different way of approaching it i would greatly appreciate the help. Thanks

2. May 7, 2006

Work you way through the problem systematically. What is Stoke's Theorem?

$$\iint_S \nabla \times \vec F \cdot d\vec S$$

What does $$d\vec S$$ mean?

$$d\vec S = \hat n \,\,dS$$

So what do you have now?

$$\iint_S \nabla \times \vec F \cdot \hat n \,\,dS$$

Now finding the curl is straightfoward, thus:

$$\iint_S \nabla \times \vec F \cdot d\vec S= \iint_S (-3,2,-2) \cdot \hat n \,\, dS$$

What's a general expression for solving a surface integral?

This one works right?
$$\iint_S f(x,y,z)\,\,dS = \iint_D f(x,y,g(x,y))\sqrt{\left(\frac{\partial z}{\partial x}\right)^2 + \left( \frac{\partial z}{\partial y} \right)^2}\,\,dA$$

Why does this work?
Well when you take the dot product of two vectors, what do you get? Yup... a scalar. And f(x,y,z) doesn't return a vector right?

So what is the unit vector? And then what happens when you take the dot product? What would f(x,y,z) equal?

Last edited: May 8, 2006
3. Jun 20, 2006

berralac

so, what exactly does d(vector)S represent? How do I get a vector to dot with curl(vector)F? My particular problem has a circle in 3-space, with z=1. The formula above is for when one is given an equation in form of z=g(x,y).

4. Jun 20, 2006

arildno

$$d\vec{S}$$ is an infinitesemal vector normal to the surface with magnitude equal to the area of the parallellogram spanned by two linearly independent tangent vectors of infinitesemal magnitudes.

5. Jun 21, 2006

HallsofIvy

Staff Emeritus
Use the "fundamental vector product": the surface of the unit sphere can be written in terms of 2 parameters, the two angles in spherical coordinates: $x= cos(\theta)sin(\phi)$, $y= sin(\theta)sin(\phi)$, $z= cos(\phi)$.
The derivatives of <x, y, z> with respect to $\theta$ and $\phi$ are the vectors
$$<-sin(\theta)sin(\phi), cos(\theta)sin(\phi), 0>$$
and
$$<cos(\theta)cos(\phi), sin(\theta)cos(\phi),-sin(\phi)>$$

The "fundamental vector product" is the cross product of those:
$$<cos(\theta)sin^2(\phi),sin(\theta)sin^2(\phi),sin(\phi)cos(\phi)>$$
(positive since it is oriented upward).

Finally, the vector differential of surface area is
$$<cos(\theta)sin^2(\phi),sin(\theta)sin^2(\phi),sin(\phi)cos(\phi)>d\theta d\phi$$