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?