Question

ques

Answer 1

How does Green's Theorem change from the original formula if we consider a clockwise line integral instead of counter-clockwise??

Answer 2

Original form
\[\oint_\limits C \phi dx+\psi dy=\iint_\limits R(\frac{\partial \psi}{\partial x}-\frac{\partial \phi}{\partial x})dxdy\]

Answer 3

@IrishBoy123

Answer 4

you get a minus answer for the integral, ie a minus area!

Answer 5

\[-\iint_\limits R (\frac{\partial \psi}{\partial x}-\frac{\partial \phi}{\partial y})dxdy?\]

Answer 6

so
\[\iint_\limits R (\frac{\partial \phi}{\partial y}-\frac{\partial \psi}{\partial x})dxdy\]
If we go clockwise ?

Answer 7

yes, its all definitional

if you go the other way you get a negative relationship between the area integral and the line integral.

so we could all agree from now on that we'll go clockwise, but we could just swicth round Greens Theorem, and life would go on

Answer 8

Is it like switching the positive and negative charges and things would still work out(it's just a convention)

Answer 9

indeed