- #1

celine

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- Homework Statement
- suppose that V is the hemisphere with |x| <= R, z>=0, F = (z, x^2+y^2, 0) in Cartesian coordinates. Verify the equality shown below by calculating both integrals and showing that they take the form (0, a, 0) in Cartesian coordinates, for some constant a to be determined.

- Relevant Equations
- It was previously proven that the first equation below should hold.

Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right.

I also tried to compute the right-hand side of the equation, letting dS = dxdy, however I'm not sure how to work with the curl between F and dS. Should I use some vector identity and rewrite this?

Thanks for your help in advance!