Green’s Theorem
Example: Apply Green’s Theorem to evaluate the work done by the force vector, F, going
completely around the square region, R, (shown below) counterclockwise.
Work = ∫ F . dr = ∫ ∫ [∂Q/∂x - ∂P/∂y] dA
C R
where F = [ y / (x2 + y2) ] i - [ x / (x2 + y2) ] j = P i + Q j
Method 1: Apply ∫ ∫ [∂Q/∂x - ∂P/∂y] dA
Now ∂Q/∂x = (x2 - y2) / (x2 + y2)2 = ∂P/∂y
Note that these partial derivatives exist throughout the region, R. Thus the double integral
over R equals zero since ∂Q/∂x = ∂P/∂y .
Method 2: Calculate the work done by evaluating the line integral directly around the
curve, C, encompassing the region R.
Click here to continue with Method 2.
Copyright © 2017 Richard C. Coddington
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