6. Consider the vector eld F = (2.173;, 3:2 + 343). (a) (a) Let C1 be the curve from (0,0) to (1,1) parametrized by r1(t) = (15,15) for 0 g t S 1. Evaluate F - dr. C1 Let C2 be the curve from (0,0) to (1,1) parametrized by r1(t) = (15,152) for O S t S 1. Evaluate F - dr. Sketch C1 and C2 on the same set of axes. C2 What do you notice about your previous two answers? What fact about F might you guess is true? 1 Now, consider f($, y) = mgy + E114. Show that Vf = F, which should conrm your guess from the previous part. You already evaluated F - dr (and F - dr) directly; now evaluate those integrals using your C1 C2 potential function f(3:, 3}). Now, let C3 be the curve consisting of a straight line segment from (U, 0) to (1,1), followed by a straight line segment from (1, 1) to (U, 1), followed by a straight line segment from (0, 1) to (U, 0) Page 2 (in that order, with that orientation). Draw a picture of C3, and evaluate F ' dr directly. (You Ca can reuse part of your work from part (a) to help you out.) Can you explain your answer to the previous part, using your answer to part (d) and/or your potential function f (3:, y)