Spherical coordinates path integral and stokes theorem

[Biffinator87]

Homework Statement

Homework Equations

The path integral equation, Stokes Theorem, the curl

The Attempt at a Solution

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sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First, does this look like the right path to the solution? I feel like I should be involving the Jacobian in the integral somewhere. Second, would the surface element be r²sin(θ) dθdφ for this and problem 2? Any help is greatly appreciated!

 

[TSny]

Looks to me like you are approaching problem 1 correctly. I'm not sure why you feel that a Jacobian [determinant (?)] should be involved here. Your surface area element is correct for problem 1. For Stokes' theorem you will need to consider the direction of the area vector for the surface element.

For the second problem, you do not have the correct surface area element. As you move around on the cone, there is no change in $\theta$. So, $d\theta = 0$ and your expression for the surface element would be zero.

Consider how you would express the area of an infinitesimal patch of the cone in the shape of a rectangle (sort of), where two of the sides are due to an infinitesimal change in $r$ and the other two sides are due to infinitesimal changes in $\phi$. You should also consider how you would express the area vector for this patch.

 

[Biffinator87]

Thanks for the help! The more I thought about it the more I understood why the Jacobian shouldn't be involved. I'll checkout that area vector and the direction of it.