In Exercises 47 and 48, prove the identity where is a simplyconnected region with

Chapter 15, Problem 47

(choose chapter or problem)

In Exercises 47 and 48 , prove the identity where R is a simply connected region with boundary C. Assume that the required partial derivatives of the scalar functions f and g are continuous. The expressions \(D_{\mathrm N}f\) and \(D_{\mathrm N}g\) are the derivatives in the direction of the outward normal vector N of C, and are defined by \(D_{\mathrm{N}} f=\nabla f \cdot \mathrm{N}\), and \(D_{\mathrm{N}} g=\nabla g \cdot \mathrm{N}\).

Green's first identity:

\(\int_{R} \int\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d A=\int_{C} f D_{\mathbf{N}} g d s\)

[Hint: Use the second alternative form of Green's Theorem and the property \(\operatorname{div}(f \mathbf{G})=f \operatorname{div} \mathbf{G}+\nabla f \cdot \mathbf{G}\).]

Text Transcription:

D_N f

D_N g

D_N f=nabla f cdot N

D_N g=nabla g cdot N

int_R int(f nabla^2 g+nabla f cdot nabla g)dA=int_C f D_Ng ds