4. Green’s Theorem states that if D is a region in the plane, and D has positive

Question

4. Green’s Theorem states that if D is a region in the plane, and D has positive orientation (i.e. counterclockwise if D has no holes), then for smooth functions P and Q we have: D P(x, y) dx + D Q(x, y) dy = D ( Q x P Q) dA . In fact this theorem is a trivial consequence of Stokes’ Theorem by defining the F in Stokes’ Theorem by: F (x, y, z) := (P(x, y), Q(x, y), 0) and observing that the upward pointing unit normal to D when considered as a surface living in IR3 is everywhere equal to (0, 0, 1). So ... Check that in this situation D F · dr can be expressed as D P(x, y) dx + D Q(x, y) dy .

Explanation / Answer

F · dr = (P(x,y)i+Q(x,y)j+0k).(dx(i)+dy(j)+dz(k)) = P(x,y)dx + Q(x,y)dy

Hence,

D F · dr =  D P(x, y) dx + D Q(x, y) dy

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