Calculus 3

Final answer only, no explanation needed

Section 13.3: Problem 1 (1 point) For each of the following vector fields F , decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, V f = F) with f(0, 0) = 0. If it is not conservative, type N. A F(I, y) = (61 - y)i + (-z + 6y)j f(I, y) = B. F(r, y) = 3yi + 4xj f(I, y) = C. F(x, y) = (3 sin y)i + (-2y + 3x cosy)j f(I, y) = Note: Your answers should be either expressions of x and y (e.g. "3xy + 2y"), or the letter "N"Section 13.3: Problem 10 (1 point) Consider the vector field IF" in the figure and the closed circular path @ oriented counter-clockwise. 4 (a) Is F . dr positive, negative, or zero? ? (b) True or False: F = grad f for some function f. Hint: use your answer to part (a). ? (c) Which of the following formulas best fits F? True False OA. F = I (12 + y?)2 OB. F -V i+ (12 + 1/2)2 (12 + 1/2)2 OC. F = -ri - VJ OD. F = -yi+ ij (Click on graph to enlarge)Section 13.3: Problem 11 (1 point) Determine whether each of the following vector fields appears to be path independent (conservative) or path dependent (not conservative). ? ? ? ? ? ? 2 (Click on a graph to enlarge it) path independent path dependent Note: You can earn 50% partial credit for 3 - 5 correct answers. Preview My Answers Submit AnswersSection 13.3: Problem 2 (1 point) Consider the vector field F (I, y, 2) = (2: + 3y)i + (42 + 3x)j + (4y + 2x)k. a) Find a function f such that F = V f and f(0, 0, 0) = 0. f (I, y, =) = b) Suppose C is any curve from (0, 0, 0) to (1, 1, 1). Use part a) to compute the line integral [F - dr.Section 13.3: Problem 3 (1 point) Consider the vector field F(I, y, 2) = zi + vi + zk. a) Find a function f such that F = V f and f(0, 0, 0) = 0. f(I, y, 2) = b) Use part a) to compute the work done by F on a particle moving along the curve C given by r(t) = (1 + 5sint)i + (1 + 4sin #)j + (1 + 2sin t)k, oct = ]. Work =Section 13.3: Problem 4 (1 point) Consider the vector field F = (x3 + 13 5ry). Compute the line integrals J F - dr and [. F - dr, where ci(t) = (t, to) and ca(#) = (t, f) for 0 1,y 1} E {(I, y) |1