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

Chapter 15, Problem 28

(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 \(D_{\mathrm{N}} f=\nabla f \cdot \mathbf{N}, \quad D_{\mathrm{N}} g=\nabla g \cdot \mathrm{N}\).

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

(Hint: Use Exercise 27 twice.)

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 - g nabla^{2} f) dV = int_S int(fD_N  g - g D_N f) dS