In this problem you will use the same vector field from Problem 2, namely F(x, y, z) (3ry, 4rz, -3y+6) (where you have already verified that div(F) = 0). Do the following: (a) Calculate a vector potential Ã 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 (z,y,z) R³, Any vector field F(z, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) Ã(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(Ã) = F 2 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(A), set it equal to F = (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 z to get a formula for N, having an integration constant f(y,z) that depends on y and z. Assume ƒ(y, z) = 0. Now you know N. Starting from the formula for R in step (b), partially integrate with respect to z 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 gly,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 r 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.

In this problem you will use the same vector field from Problem 2, namely F(x, y, z) (3ry, 4rz, -3y+6) (where you have already verified that div(F) = 0). Do the following: (a) Calculate a vector potential Ã 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 (z,y,z) R³, Any vector field F(z, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) Ã(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(Ã) = F 2 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(A), set it equal to F = (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 z to get a formula for N, having an integration constant f(y,z) that depends on y and z. Assume ƒ(y, z) = 0. Now you know N. Starting from the formula for R in step (b), partially integrate with respect to z 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 gly,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 r 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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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