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

C R

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.

Work = ∫ F . dr = ∫ ∫ [∂Q/∂x - ∂P/∂y] dA

C R

Click here to continue with Method 2.

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