Question : Use Gauss theorem to evaluate the surface integral\[\iint_{\gamma}{\left(x^2+y+z\right)dS\ }\]Where γ is the surface of the ball\[ x^2+y^2+z^2\le1\] Solution : Unit Normal vector to surface \[\gamma\ =x^2+y^2+z^2-1\]Is given as\[\hat{n}=\frac{\nabla\gamma}{\left|\nabla\gamma\right|}\] \[=\frac{\left(2x\hat{i}+2y\hat{j}+2z\hat{k}\right)}{\sqrt{4x^2+4y^2+4z^2}}\]\[=\frac{\left(x\hat{i}+y\hat{j}+z\hat{k}\right)}{\sqrt{x^2+y^2+z^2}}\]Since x^2+y^2+z^2=1\[=>\ \hat{n}=x\hat{i}+y\hat{j}+z\hat{k}\] Let a vector field F=(f1,f2,f3) such that\[\vec{F}.\hat{n}=x^2+y+z\]\[=>\left(f_1\hat{i}+f_2\hat{j}+f_3\hat{k}\right).\left(x\hat{i}+y\hat{j}+z\hat{k}\right)=x^2+y+z\]\[=>{xf}_1+yf_2+zf_3=x^2+y+z\]\[=>f_1=x,\…
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