Multivariate Calculus Problem

A) Using the formula for $E$ we compute 
\[\iint_{S^2_{R_p}}E\cdot dA=\iint_{S^2_{R_p}}\frac{R_p-a}{R_p^2}dA=\frac{R_p-a}{R_p^2}\iint_{S^2_{R_p}}dA\]\[=\frac{R_p-a}{R_p^2}\times 4\pi R_p^2=4\pi(R_p-a).\]

Note that $\frac{R_p-a}{R_p^2}$ is constant on the sphere $S^2_{R_p}$ and that is why we can bring it outside of the integral. 

B) We have 
\[\iiint_{S_{aR_P}}\rho dV=\iiint_{S_{aR_P}}\frac{1}{r^2} dV=\int_a^{R_P}\int_{S^2_r}ds \frac{1}{r^2}dr\] 
\[=\int_a^{R_P}4\pi r^2 \cdot \frac{1}{r^2}dr=4\pi\int_a^{R_P}dr=4\pi(R_p-a).\]

Here we have used Coarea formula in polar coordinates (see Theorem 1.1 here )

C- Using the formula for divergence in spherical coordinates (see Wikipedia) we have 
\[\nabla E=\frac{1}{R_p^2}\frac{\partial (R_p^2 E)}{\partial R_P}=\frac{1}{R_p^2}\frac{\partial (R_p-a)}{\partial R_P}=\frac{1}{R_p^2}=\rho.\]

It follows from Parts A, B, and C that 
\[\iint_{S^2_{R_p}}E\cdot dA=\iiint_{S_{aR_P}}\rho dV=\iiint_{S_{aR_P}}\nabla \cdot E dV.\]

This means that the electric flux through $S^2_{R_P}$ is equal to the total electric charge enclosed by this surface. This is indeed Gauss’s Law for Electric Fields