1. Let F = -yi + xj and let C be the unit circle oriented counterclockwise. (a) Show that F has a constant magnitude of 1 on C. (b) Show that F is always tangent to the circle C. (c) Show that F - dr =length of C. (d) Without calculating the curl, determine whether F is path-independent. Explain your reasoning. 2. Consider the vector field F(x, y) = evi+ (xe" + e²)j + ye*k. %3D (a) Show that F is conservative by determining a function f such that F = Vf. (b) Find a parameterization of C, the line segment from (0, 2, 0) to (4, 0, 3). (c) Set up the line integral f. F - dr, where C is the line segment from (0, 2, 0) to (4,0, 3). (d) Evaluate fe F - dr using whatever method you think is best. 3. Let i = V(r? + y?). Consider the path C which is a line between any two of the following points: (0,0), (-5,0), (0, 6), (0, –6), (5,4), (-3, –5). Suppose you want to choose the path C in order to maximize ở - dr. What point should be the start of C? What point should be the end of C? Explain your answer.

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