- State the divergence theorem. Describe the regions over which the integrations are carried out and the quantities that are being integrated.
Consider the vector field $$\mathbf{V}=\frac{3y\mathbf{i}+2xz\mathbf{j}-z\mathbf{k}}{\sqrt{x^2+y^2+z^2}}.$$ Find the outward flux across the boundary of the hemispherical volume bounded from below by the spherical surface $x^2+y^2+z^2=R^2$ (for $z<0$) and from above by the $x-y$ plane.
Use the divergence theorem to verify the result.
My attempt at the question involved me using the divergence theorem as follows:
$${\large\bigcirc}\kern{-1.55 em}\iint\limits_{S}\vec{\mathbf{F}}\cdot d\vec{\mathbf{S}}=\iiint\limits_{D}\text{div}(\vec{\mathbf{F}})\,dV$$
By integrating using spherical coordinates it seems to suggest the answer is $-\frac{2}{3}\pi R^2$. We would expect the same for the LHS. My calculation for the flat section of the hemisphere is zero while the curved surface is $-\pi R^2$.
Therefore it appears that the RHS and LHS do not agree but I'm unsure what went wrong. Any advice or help would be appreciated.