anonymous
  • anonymous
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Mathematics
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chestercat
  • chestercat
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anonymous
  • anonymous
How does Green's Theorem change from the original formula if we consider a clockwise line integral instead of counter-clockwise??
anonymous
  • anonymous
Original form \[\oint_\limits C \phi dx+\psi dy=\iint_\limits R(\frac{\partial \psi}{\partial x}-\frac{\partial \phi}{\partial x})dxdy\]
anonymous
  • anonymous
@IrishBoy123

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IrishBoy123
  • IrishBoy123
you get a minus answer for the integral, ie a minus area!
anonymous
  • anonymous
\[-\iint_\limits R (\frac{\partial \psi}{\partial x}-\frac{\partial \phi}{\partial y})dxdy?\]
anonymous
  • anonymous
so \[\iint_\limits R (\frac{\partial \phi}{\partial y}-\frac{\partial \psi}{\partial x})dxdy\] If we go clockwise ?
IrishBoy123
  • IrishBoy123
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
anonymous
  • anonymous
Is it like switching the positive and negative charges and things would still work out(it's just a convention)
IrishBoy123
  • IrishBoy123
indeed

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