Question 26.3: If Φ = x³y + xy² + 3y find (a) ∇Φ (b) ∇Φ|(0,0,0) (c) |∇Φ| at......

(a) If Φ = x³y + xy² + 3y then

\frac{\partial\phi}{\partial x}=3x^{2}y+y^{2}

{\frac{\partial\phi}{\partial y}}=x^{3}+2x y+3

\frac{\partial\phi}{\partial{{{z}}}}=0

so that

\nabla\phi=(3x^{2}y+y^{2})\mathbf{i}+(x^{3}+2x y+3)\mathbf{j}+0\mathbf{k}

(b) At  (0,0,0),\nabla\phi=0\mathbf{i}+3\mathbf{j}+0\mathbf{k}=3\mathbf{j}.

(c)  At   (1,1,1),\nabla\phi=(3\times1^{2}\times1+1)\mathbf{i}+(1^{3}+2\times1\times1+3)\mathbf{j}+0\mathbf{k}=4\mathbf{i}+6\mathbf{j}+0\mathbf{k}  so that  |\nabla\phi|{\mathrm{~at~}}(1,1,1)  is equal to  {\sqrt{4^{2}+6^{2}}}={\sqrt{52}}.