Calculating Flux Using Stoke's Theorem for a Spherical Surface

[sunnyday11]

Homework Statement

F(x,y,z) = zy i + z k

and the surface S defined by x² + y² + z² = 4, z$$\geq\sqrt{3}$$

Homework Equations

The Attempt at a Solution

Using line integral method, I got -$$\sqrt{3}$$$$\pi$$

But using the flux of curl method, I got curl = y j - z k
Then I change surface integral to double integral using z=$$\sqrt{4 - x² - y²}$$
and parameterization r(t) = rcost i + rsint j.

$$\int$$ $$^{0}_{1}$$$$\int$$ $$^{0}_{2pi}$$ ( r³sin²t / $$\sqrt{4-r²}$$ - $$\sqrt{4-r²}$$ ) dt dr


Then I arrive at $$\int$$ $$^{0}_{1}$$ ( r³/$$\sqrt{4-r²}$$ - $$\sqrt{4-r²}$$ ) dr

and I got stuck.

Thank you very much!

 

[vela]

Homework Statement

F(x,y,z) = zy i + z k

and the surface S defined by x² + y² + z² = 4, z$$\geq\sqrt{3}$$

Homework Equations

The Attempt at a Solution

Using line integral method, I got -$$\sqrt{3}\pi$$

But using the flux of curl method, I got curl = y j - z k

Good up to here, but I can't follow what you did below.

Then I change surface integral to double integral using z=$$\sqrt{4 - x² - y²}$$
and parameterization r(t) = rcost i + rsint j.

$$\int$$ $$^{0}_{1}$$$$\int$$ $$^{0}_{2pi}$$ ( r³sin²t / $$\sqrt{4-r²}$$ - $$\sqrt{4-r²}$$ ) dt dr


Then I arrive at $$\int$$ $$^{0}_{1}$$ ( r³/$$\sqrt{4-r²}$$ - $$\sqrt{4-r²}$$ ) dr

and I got stuck.

Thank you very much!

So you want to evaluate

$$\int (\nabla\times \mathbf{F})\cdot d\mathbf{S} = \int (y \hat{\mathbf{j}} - z \hat{\mathbf{k}})\cdot d\mathbf{S}$$

Since the surface is a piece of a sphere, I suggest you switch to spherical coordinates. What are the magnitude and direction of the area element dS?