In this problem you will use the same vector field from Problem 2, namely F(x, y, z) = (3xy, 4xz, -3yz+6) (where you have already verified that div(F) = 0). Do the following: (a) Calculate a vector potential Ā for F. (b) Check your answer by verifying that curl(A) = F. For (a) you can use the step-by-step method from class. Here is a quick review of that method; you can also consult class notes. Consider a C¹ vector field defined for all (x, y, z) = R³, F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) Any vector field A(x, y, z) = (L(x, y, z), M(x, y, z), N(x, y, z)) which is a solution to the vector differential equation curl(A) = F is called a vector potential for F. (We know from class or from Briggs calculus that div(curl(A)) 0, so only a vector field with divergence 0 = can have a vector potential; for this problem, you have already calculated that div(F) = 0 in problem 2). Here is a procedure to compute a vector potential: = Starting from A (L, M, N) where L, M, N are unknown, write the formula for curl(Ã), set it equal to ♬ = (P,Q,R), and separate into three component formulas: one for P, one for Q, and one for R. • Assume L = 0, and use assumption that to simplify the component formulas for P, Q and R. Now you know L. Starting from the formula for Q in step (b), partially integrate with respect to x to get a formula for N, having an integration constant f(y, z) that depends on y and z. Assume f(y, z) = 0. Now you know N. Starting from the formula for R in step (b), partially integrate with respect to x to get a formula for M. You should again have an integration constant g(y, z) that again depends on y and z. This time do NOT assume that g(y, z) is equal to 0. Now you partly know M, except you don't yet have a formula for g(y, z). Starting with the formula for P in step (b), -plug in your formulas for N in step (c) -plug in your formula for M in step (d). - Simplify and solve for g(y, z). (There should be no dependence on x in your formula for g(y, z); if there is, you've done something wrong, so go back and check through your work). Now you know M completely. ⚫ Putting it altogether, you've found a formula for a vector potential A = (L, M, N) of the vector field F.