$S = \lbrace (x,y,z) \: | \: (x + 4)^2 + (y-5)^2 + (z + 1)^2 = 9 \rbrace$
Find the flux of $\vec{F} = 2 x^{2} \,\hat{\imath} - 3 y^{2} \,\hat{\jmath} - 2 z^2 \, \hat{k}$ in $(-4,5,-1)$.
My attempt:
Using Gauss theorem $$ \int_S \vec{F} \cdot \hat{n} \, dS = \iiint_V \nabla \cdot \vec{F} dx dy dz$$
$\nabla \cdot \vec{F} dx dy dz=4x-6z-4z=4(-4)-6(5)-4(-1)=-42$.
Therefore,$\iiint_V \nabla \cdot \vec{F} dx dy dz= \iiint_V-42 dx dy dz$.
Denote $x = ρ\cosθ\sinϕ,y = ρ\sinθ\sinϕ,z = ρ\cosϕ$.
$∫_0^\pi∫_0^{2\pi}∫_0^3-42\cdotρ^2\sin(ϕ)dρdθd=1512\pi $
My solution is wrong , can't find out the problem, appreciate any help.