Question

Use Green's first identity to show ∭|∇f|²dV = ∬(df/dn) dS.

Use Green's first identity to show ∭|∇f|²dV = ∬(df/dn) dS.

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Answer #1

Answer: Let be a closed surface and be the region bounded by the surface . Let be the normal vector to the surface . Then, it follows from the definition of the directional derivative that

It also follows from the definition of the gradient and divergence that

Now, by Gauss' divergence theorem (Green's first identity), it follows that

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