1. Consider the vector field F in the plane given by F(x, y) = (-r’y)i + (xy?)j. Let C, denote the smooth oriented path

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1 Consider The Vector Field F In The Plane Given By F X Y R Y I Xy J Let C Denote The Smooth Oriented Path 1 (61.27 KiB) Viewed 21 times

1. Consider the vector field F in the plane given by F(x, y) = (-r’y)i + (xy?)j. Let C, denote the smooth oriented path counterclockwise once around the circle parame- terized by C.: r(t) = a cos(t)i +a sin(t)j, Ost<2n, where a > 0 is a constant determining the radius of the circle. Let R. denote the corre- sponding disc of radius a consisting of all points (x,y) satisfying Ra: r?+ y <a The purpose of this exercise is to explicitly confirm that Green's Theorem holds for the above. (a) Find the work done by F around C, by explicitly computing the line integral . F. Tds. Your answer will depend on a. Show all your work including Change of Vari- ables. You may need to review how to compute some trigonometric integrals. (b) Find the curl of F. Then explicitly compute the double integral of the curl over the region R,, i.e. Il curl FdA. You will want to convert to polar coordinates to do this. Warning: don't confuse the polar variable r with the constant radius of the region's boundary a. Remember that dA = r dr de when converting integrals to polar form. (c) Observe that the results agree; this is Green's Theorem. Is the field F conservative on the plane? Explain. Figure 1: Exercise #1