E field between 2 wrinkled spheres, radial symmetry?

[mathnerd15]

\oint E\cdot dA=|E|\int_{0}^{2\pi}\int_{0}^{\pi}(1+1/2sin6\theta\sin5\phi)^2sin\phi d\phi d\theta =|E|\int_{0}^{2\pi}(\frac{25}{99}sin^2(6\theta)+2) d\theta =|E|\frac{421\pi}{99}=
\frac{\rho_{q}}{\varepsilon o}\int_{0}^{2\pi }\int_{0}^{\pi }\int_{0}^{(1+\frac{1}{2}sin6\theta sin5\phi )}\rho^2sin(\phi )d\rho d\phi d\theta=
\frac{\rho_{q}}{\varepsilon o} \int_{0}^{2\pi }\int_{0}^{\pi } \frac{1}{198}(157-25cos12\theta )d\phi d\theta= \frac{157\pi\rho_{q}}{99\varepsilon o }... E=\rho_{q}\frac{157}{421\varepsilon o}...Er_{1}-Er_{2} = \frac{\rho_{q}d157}{421\varepsilon o}

is a wrinkled sphere also an S2 object topologically though maybe different Euler characteristic? is this calculation correct since the field between 2 regular spheres has a radial term with symmetry? these spheres I read are used to model tumors

thanks very much!

 

[mathnerd15]

this seems to be a good approach to design your own problems that are perhaps variations on standard ones, they do this at Princeton for instance