?Green's Formula Write Gauss' Formula of Exercise 48 in two dimensions-that is, where

Chapter 14, Problem 1056

(choose chapter or problem)

Green's Formula Write Gauss' Formula of Exercise 48 in two dimensions-that is, where \(\mathbf{F}=\langle f, g\rangle\), D is a plane region R and C is the boundary of R. Show that the result is Green's Formula:

 \(\iint_{R} u\left(f_{x}+g_{y}\right) d A=\oint_{C} u(\mathbf{F} \cdot \mathbf{n}) d s-\iint_{R}\left(f u_{x}+g u_{y}\right) d A\)

Show that with II = I . one form of Green's Theorem appears. Which form of Green's Theorem is it?

Text Transcription:

F = langle f, g rangle

iint_R u(f_x + g_y) dA = oint_C u(F cdot n)ds - iint_R(fu_x + gu_y)dA