Math 53 Quiz 811 April 2008GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/Name:Time (circle one): 12:10 - 1:00 3:10 - 4:00Please use extra paper as necessary. For each part, partial credit will be assigned based on correctwork (you do need to show some work, enough so that I know how you solved the problem). Pleasesimplify and box your answers.Consider the vector field:~v(x, y) = y~ı + x~a. (1 pt) Which of the following is a graph of ~v?(Note: My graphing calculator displays only the direction of vectors, normalizing each to aunit vector, and displaying the magnitude as a color. But this is a black-and-white printer.)b. (5 pt) Consider the path ~r(t) = (x(t), y(t)) = (1 + t, 3 − t) as t ranges from 0 to 1. Computedirectly the line integralZpath~v(~r) · d~r(I.e. set up the integral in terms of t and use the methods of 13.2, but not 13.3.)c. (4 pt) Is there a function F (x, y) so that ~v(x, y) =~∇F ? If so, find such a function, andevaluate F (2, 2) − F (1, 3). If not, explain why
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