Question of Green's Theorem

1. Apr 19, 2013

yungman

This is a copy of the book:
[PLAIN]http://i38.tinypic.com/20faqnc.jpg[/PLAIN]

I know the derivation part, I just want to see whether I understand why the -ve sign of $-\frac {\partial f}{\partial y}dA$ in a more common sense way.

From looking at the graph for type I region, $g_2(x)$ is above $g_1(x)$ referenced to y axis. So the integral has to be $g_2(x)-g_1(x)$. BUT the orientation of curve of $g_2(x)$ is from b to a. So if we want to integrate from a to b, we need to put a -ve sign.

From the type II region, $h_2(x)$ is above $h_1(x)$ referenced to x axis. So the integral has to be $h_2(x)-h_1(x)$. The orientation of curve of $h_2(x)$ is from c to d. So if we want to integrate from c to d, it would be +ve sign.

Am I getting it right?

Bottom line is the sign depends on the direction of the higher value function of the two ( ie. $g_2(t)≥g_1(t)$). If the direction is from high value to low value, then the sign has to be change to make it from low to high ( ie. $g_2(t)$ oriented from b to a. So sign needed to be change to integrate from a to b).

Last edited: Apr 19, 2013
2. Apr 20, 2013