Stokes' Theorem

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
Gold Member
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.

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

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
Gold Member
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.

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
Gold Member
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.