Test review
MATH 3433 - CALCULUS III ONLINE (7) [15 Points] Let F = (sin(x) +2y+z, 3z+cos(y), 4x -ez) and C' be the positively Exam 5 oriented, closed, triangular curve with vertices at (2, 0, 0), (0, 2, 0), and (0, 0, 4). . Find the value of Jo, F . T ds, where Ci is the first part of the curve (from Name: Date: (2, 0, 0) to (0, 2, 0)). Use blank paper to complete this test - make sure problems are all labeled clearly and submitted in order. You must show all your work in a clear format to receive full credit . Use Stokes's Theorem to evaluate fo F . T ds. on each problem. Remember to simplify all answers. . Determine whether calculating the line integral or the surface integral is eas- (1) [5 Points] Compare and contrast Green's theorem and Stokes' theorem. What ier and explain why. Use this to motivate whether or not applying Stokes' do they have in common? What are the key differences in the applications of the Theorem to calculate fo F . T ds is warranted. two theorems? (2) [5 Points] Explain how to turn a line integral of the form ( P dr + Q dy + R dz (8) [15 Points] Let F = (2x + yz, xz - y, x y + 32) be a vector field defined over the volume E = [0, 3] x [1, 2] x [-1, 1] and let S be the 6 sides which form the into a single integral fo f ( t ) dt . boundary of E . (3) [5 Points Each] Explain how to do each of the following. . Find the flux of F through the top of E. Write the surface integral that gives . How do you find the area of a parallelogram formed by two vectors? the flux and then evaluate the integral. . How do you determine if two vectors are parallel? . Use the Divergence Theorem to find the flux of F through all of S. . How do you determine if two vectors are orthogonal? . Determine whether calculating the surface integral or the volume integral is easier and explain why. Use this to motivate whether or not applying the . How can you use derivatives to determine if a 2D vector field (P(x, y), Q(x, y)) Divergence Theorem to calculate the flux of F across all of S is warranted . is conservative? . What can you calculate to determine if a 3D vector field (P(x, y, z), Q(x, y, z), R(x, y, z)) is conservative? . How can you tell if a vector field is really the curl of another vector field? (4) [10 Points] Evaluate So F . dr where F = (7yz + 4, 7xz + In(x), 7xy) and C is a curve which has initial point (1, 2, 3) and terminal point (2, 0, -3). (5) [10 Points] Evaluate J. 3xy dx - xz dy + 2 dz where C is the curve formed by traveling along the unit circle from (0, 1, 1) to (1, 0, 1) and then traveling along the straight line from (1, 0, 1) to (2, 1, 1). (6) [10 Points] Let C be the unit circle and define functions P(x, y) and Q(x, y) as follows: P (x, y ) = - Q (x, y) = (x2 + 12 . For what (x, y) values is the vector field (P(x, y), Q(x, y)) a conservative vector field? Justify your answer. . Can Green's theorem be applied to evaluate fo Pdr + Qdy? Explain why or why not
