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By definition,\nabla g=({\frac  {\partial g}{\partial x}})i+({\frac  {\partial g}{\partial y}})j+({\frac  {\partial g}{\partial z}})k\,

Hence f\nabla g=[f{\frac  {\partial g}{\partial x}}]i+[f{\frac  {\partial g}{\partial y}}]j+[f{\frac  {\partial g}{\partial z}}]k\,

and so by definition of divergence, we get

{\mathrm  {div}}(f\nabla g)={\frac  {\partial }{\partial x}}[f{\frac  {\partial g}{\partial x}}]+{\frac  {\partial }{\partial y}}[f{\frac  {\partial g}{\partial y}}]+{\frac  {\partial }{\partial z}}[f{\frac  {\partial g}{\partial z}}]\,

={\frac  {\partial f}{\partial x}}{\frac  {\partial g}{\partial x}}+f{\frac  {\partial ^{2}g}{\partial x^{2}}}+{\frac  {\partial f}{\partial y}}{\frac  {\partial g}{\partial y}}+f{\frac  {\partial ^{2}g}{\partial y^{2}}}+{\frac  {\partial f}{\partial z}}{\frac  {\partial g}{\partial z}}+f{\frac  {\partial ^{2}g}{\partial z^{2}}}\,

=f[{\frac  {\partial ^{2}g}{\partial x^{2}}}+{\frac  {\partial ^{2}g}{\partial y^{2}}}+{\frac  {\partial ^{2}g}{\partial z^{2}}}]+[{\frac  {\partial f}{\partial x}}{\frac  {\partial g}{\partial x}}+{\frac  {\partial f}{\partial y}}{\frac  {\partial g}{\partial y}}+{\frac  {\partial f}{\partial z}}{\frac  {\partial g}{\partial z}}]\,

=f[{\frac  {\partial ^{2}}{\partial x^{2}}}+{\frac  {\partial ^{2}}{\partial y^{2}}}+{\frac  {\partial ^{2}}{\partial z^{2}}}]g+[{\frac  {\partial f}{\partial x}}i+{\frac  {\partial f}{\partial y}}j+{\frac  {\partial f}{\partial z}}k]\cdot [{\frac  {\partial g}{\partial x}}i+{\frac  {\partial g}{\partial y}}j+{\frac  {\partial g}{\partial z}}k]\,

=f\nabla ^{2}g+\nabla f\cdot \nabla g\,

Thus,{\mathrm  {div}}(f\nabla g)=f\nabla ^{2}g+\nabla f\times \nabla g\,

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