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MATH 3433 - CALCULUS III ONLINE Exam 5 Name: Date: 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 on each problem. Remember to simplify all answers. (1) [5 Points] Compare and contrast Green's theorem and Stokes' theorem. What do they have in common? What are the key differences in the applications of the two theorems? (2) [5 Points] Explain how to turn a line integral of the form JoPdz+Qdy+Rdz into a single integral [ f() dt. (3) [5 Points Each] Explain how to do each of the following. o How do you find the area of a parallelogram formed by two vectors? e How do you determine if two vectors are parallel? e How do you determine if two vectors are orthogonal? o How can you use derivatives to determine if a 2D vector field (P(z,y), Q(z,v)) is conservative? e What can you calculate to determine if a 3D vector field (P(z,y,2), Q(z, v, 2), R(z,y, 2)) is conservative? o How can you tell if a vector field is really the curl of another vector field? o (4) [10 Points] Evaluate ._,nw, -dr where F = (Tyz + L, 7zz + In(z), 7zy) and C is 2 curve which has initial point (1,2,3) and terminal point (2,0, 3). (5) [10 Points] Evaluate [, 3zy dz zz 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(z,y) and Q(z,y) as follows: z Yy ) T,Y) = e Q)= = e For what (z,y) values is the vector field (P(z,y), Q(z,y)) a conservative vector field? Justify your answer. o Can Green's theorem be applied to evaluate , Pdz + Qdy? Explain why or why not. P(z,y) = a (7) _Hm Points] Let F' = (sin(z) +2y+2, 3z+cos(y), 4z e?) and C be the positively oriented, closed, triangular curve with vertices at (2,0,0), (0,2,0), and (0,0,4). i - o e Find the value of \\.9 F-T ds, where C) is the first part of the curve (from (2,0,0) to (0,2,0)). e Use Stokes's Theorem to evaluate %Q w . m.; ds. .UmnmnBEm whether calculating the line integral or the surface integral is eas- ier and explain why. Use this to motivate whether or not applying Stokes' Theorem to calculate .mo M.V_ . m., ds is warranted. = (8) [15 Points| Let F' = (2z + yz,2z y,z%y + 3z) 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 boundary of E. 5 =) o Find the flux of F' through the top of E. Write the surface integral that gives the flux and then evaluate the integral. e Use the Divergence Theorem to find the flux of N_ through all of S. Ummmdium whether calculating the surface integral or the volume integral is easier and explain why. Use this to motivate whether or not applying the Divergence Theorem to calculate the flux of w,. across all of S is warranted