# Stokes' Theorem

JaysFan31
Here's my problem:
Take u=(x^3)+(y^3)+(z^2) and v=x+y+z and evaluate the surface integral
double integral of grad(u) x grad(v) ndS where x is the cross product and between the cross product and the ndS there should be a dot product sign. The region S is the hemisphere x^2+y^2+z^2=1 with z greater than or equal to 0 and n has non-negative k component.

Here's my work:

curl F * k = (-6x+6y)
Thus I=double integral of curl F * ndS = double integral of curl F *k dA = double integral of (-6x+6y)dA.

I used polar coordinates:
-6 integral from 0 to 2pi integral from 0 to 1 (rcost-rsint) rdrdt. However, this integral equals zero. Is this right?

Is this flux zero?

cristo
Staff Emeritus
Here's my problem:
Take u=(x^3)+(y^3)+(z^2) and v=x+y+z and evaluate the surface integral
double integral of grad(u) x grad(v) ndS where x is the cross product and between the cross product and the ndS there should be a dot product sign.
Is this what you mean: $$\int\int\nabla u \times\nabla v \cdot \bold{n}dS$$?

The region S is the hemisphere x^2+y^2+z^2=1 with z greater than or equal to 0 and n has non-negative k component.

Here's my work:
Where does the minus sign in front of the j component come from? $$\nabla u=\frac{\partial u}{\partial x}\bold{i}+\frac{\partial u}{\partial y}\bold{j}+\frac{\partial u}{\partial z}\bold{k}$$
These look correct with the amended form for grad(u)

curl F * k = (-6x+6y)
Thus I=double integral of curl F * ndS = double integral of curl F *k dA = double integral of (-6x+6y)dA.
What's F? You should define anything you introduce. I imagine you've used Stokes' Theorem here, but where?

[As an aside, it would be helpful if you could learn LaTex; it's very simple to learn! Click on one of the images in my text, and follow the link to the tutorial]

Last edited:
Dick
Homework Helper
I think I would agree the total flux is zero, but I don't see how you are getting it. What does the stated problem have to do with Stokes Theorem? Where did the curl come from? What is F?

JaysFan31
Yes, the integral that cristo wrote is what I'm looking for.

I made a mistake. u should equal (x^3)-(y^3)+(z^2). That's why I have a negative j component.

If I let F equal the above, isn't this correct?

Dick