Proof In Exercises 27 and 28, prove the identity, assuming that and N meet the

Chapter 15, Problem 27

(choose chapter or problem)

In Exercises 27 and 28, prove the identity, assuming that Q, S, and N meet the conditions of the Divergence Theorem and that the required partial derivatives of the scalar functions f and g are continuous. The expressions \(D_{N} f\) and \(D_{N} g\) are the derivatives in the direction of the vector N and are defined by \(D_{\mathrm{N}} f=\nabla f \cdot \mathbf{N}, \quad D_{\mathrm{N}} g=\nabla g \cdot \mathrm{N}\).

\(\iiint_{Q}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V=\int_{S} \int f D_{\mathrm{N}} g d S\)

[Hint: Use div \((f \mathbf{G})=f\) div \(\mathbf{G}+\nabla f \cdot \mathbf{G}\).]

Text Transcription:

D_N f

D_N g

D_N f = nabla f cdot N, D_N g = nabla g cdot N

iiint_Q (f nabla^{2} g + nabla f cdot nabla g) dV = int_S int f D_N g dS

div (f G) = f div G + nabla f cdot G