Homework Help: Compute circulation of vector around the contour

1. May 11, 2013

princessme

1. The problem statement, all variables and given/known data

Compute the circulation of the vector a = yi+x2j - zk around the contour L: {x2 +y2 = 4; z = 3}, a) directly and b) via the Stokes Theorem.
Plot the contour and show its orientation.

2. Relevant equations

Stokes theorem is $\oint$F.dr = $\int$∇ X F . n dS

3. The attempt at a solution

For (a) which is solving it directly, i used the left side of the equation $\oint$F.dr and I obtained 0 as my answer. Having problem with part b. What should my ∇f be? Is it 2x i + 2y j + 0 k?

2. May 11, 2013

HallsofIvy

No, it isn't. Don't you know how $nabla\times \vec{F}$ is defined? If $\vec{F}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$ then
$$\nabla\times \vec{F}= \left(\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z}\right)\vec{i}-\left(\frac{\partial h}{\partial x}- \frac{\partial f}{\partial z}\right)\vec{j}+ \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right)\vec{k}$$

3. May 11, 2013

princessme

Yes I know that. But to obtain a unit normal vector n, i need to first find $\frac{∇f}{|∇f|}$ , isn't it?

Anyway, I managed to find a mistake I made for the left hand side, and obtained my answer as -4pi. Mind guiding me through for the right hand side?

4. May 17, 2013

Himmeltan

How did you get -4pi? Also, did you manage to finish part b?

5. May 18, 2013

I like Serena

No.

The unit normal vector $\mathbf n$ is normal to the surface inside the contour.
The obvious choice for that surface is in the plane z=3, bounded by a circle.
The unit normal vector to that plane is $\mathbf n = \mathbf k$.

So first you have to determine $\nabla \times \mathbf a$, and then you have to take the dot product with $\mathbf n = \mathbf k$.

6. May 21, 2013

dan38

How do you know it's "k" and not negative "k"

7. May 22, 2013

I like Serena

You can choose the direction.
The consequence is that for the contour integral you have to follow the contour according to the right hand rule.