1. (Briggs section 17.3 problem 31) On R² let (x, y) = xy F(x,y) = √(x, y) (t) = (cost, sint) for 0≤t≤T Note that (t) is a parameterization of the upper unit semicircle. Calculate Fd in two ways: (a) First calculate a formula for F(x, y), and then evaluate the path integral directly. (b) Use the fundamental theorem of calculus for path integrals. 2. Suppose that F is any force field and √(t) is any constant path. Calculate SF.d7. 3. Let be the boundary of the square [0, 1] × [0, 1] parameterized in the coun- terclockwise direction, and broken into a concatenation of four straight line segments Y1 * 2 * 3 * 4 as follows: Y₁ is the line segment from (0,0) to (1,0) Y2 is the line segment form (1,0) to (1, 1) Y3 is the line segment from (1, 1) to (0, 1) ⚫ 4 is the line segment from (0, 1) to (0,0) Answer the following questions about the force field F(x, y) = (x² — y², 2xy) (a) Compute the amount of work done by ♬ in moving a particle in the counterclockwise direction around the square [0,1] × [0, 1] along the path from (0,0) and back to (0, 0). (b) Is F conservative? (Hint: Can you think of another path from (0,0) to (0,0) whose work integral is easy to calculate?)

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