Verifying Stokes' Theorem for a Hemispherical Cap

[yitriana]

Homework Statement

Suppose we want to verify Stokes' theorem for a vector field F = <y, -x, 2z + 3> (in cartesian basis vectors), where the surface is the hemispherical cap +sqrt(a^2 - x^2 - y^2)

The Attempt at a Solution

Why is it that if I substitute spherical coordinates x = asinθcosΦ, y = asinθsinΦ, z = acosθ, into F, and then take the curl where del = <d/dr, d/dθ, d/dΦ> (note that r = a in this case, so if the term involves only a, the d/dr of that term is 0), when I do double integral over curl F * dS with θ and Φ as parameters, I get 0?

However, if I first take the curl of F where del = <d/dx, d/dy, d/dz> and THEN substitute spherical coordinates into the curl F, and dot with dS, and do double integral, I get the right answer ( i know the "right answer" because of the simplicity of the line integral).

Why can I not substitute spherical coordinates into F and then take the curl where del = <d/dr, d/dθ, d/dΦ>?

Why must I take the curl F where del = <d/dx, d/dy, d/dz> and then substitute spherical coordinates?

 

[Pengwuino]

The curl in spherical coordinates is not something like $$\frac{\partial }{{\partial r}},\frac{\partial }{{\partial \theta }},\frac{\partial }{{\partial \varphi }}$$

It's more complicated then that. Whatever text you are using should have the curl in spherical coordinates. It's quite long.

Edit: Ah, thank you wikipedia: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates you're looking for the curl in spherical coordinates.

 

[yitriana]

Oh sorry. I was parametrizing in spherical coordinates but not necessarily using spherical basis vectors, which explains why I took the curl in cartesian coordinates (and did not need scale factors)

 

[Pengwuino]

hmm? You didn't attempt to take the curl in cartesian though, you attempted to take them in spherical.

 

[yitriana]

I used spherical coordinate parameters but wrote in terms of x, y and z (cartesian coordinates).

Ah. I figured out my mistake now. Since I did not rewrite F in spherical coordinates, I *cannot* take the curl with del as <d/dr, d/dtheta, d/dphi>--that would not work since F was expressed in cartesian coordinates.

Thanks.

 

[Pengwuino]

Well, remember, even if you put F in terms of spherical coordinates in the cartesian basis, the curl isn't simply $$\frac{\partial }{{\partial r}}\hat x + \frac{\partial }{{\partial \theta }}\hat y + \frac{\partial }{{\partial \phi }}\hat z$$ either.

 

[yitriana]

is it even possible to compute the curl of F of in terms of spherical coordinates in the cartesian basis, or would you have to convert F to spherical basis if F is in terms of spherical coordinates to compute the curl?

(of course, the curl of F could be computed in cartesian basis in terms of cartesian coordinates, but just wondering)

 

[Pengwuino]

Yes it's possible but kinda ugly I can imagine since you'd have to determine things like $$\hat x \cdot \hat \theta$$ that'll add to your derivatives. It's best to simply convert everything to the proper basis and parametrization for what you want to do.