VC5.30

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Let V be the volume enclosed by S.By definition,

{\mathrm  {div}}A={\frac  {\partial }{\partial x}}(2x+3z)+{\frac  {\partial }{\partial y}}(-xz-y)+{\frac  {\partial }{\partial z}}(y^{2}+2z)=2-1+2=3\,

Hence,by the divergence theorm,we have

\iint _{S}A\cdot ndS=\iiint _{V}{\mathrm  {div}}AdV=\iiint _{V}3dV=3V=3\times 36\pi \,

Since,V=volume of given sphere of radius 3={\frac  {4}{3}}\pi (3)^{3}=36\pi \,

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