4. (8 points) Let F = (P,Q) be the vector field defined by P(x, y) =...

Question:

4. (8 points) Let F = (P,Q) be the vector field defined by P(x, y) = +y (x, y) = (0,0) 22+y2; 0, (x, y) = (0,0) Q(x, y) = -+y 22 +y2; (x, y) = (0,0) 10, (x, y) = (0,0). (a) (3 points) Show that F is a gradient vector field in R2 \ {y=0}. (b) (4 points) Letting D= {22020 + y2020 < 1}, compute the line integral Sap P dx +Q dy in the counter-clockwise direction. (c) (1 point) Does your calculation in part (b) violate Green's theorem? Explain.

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