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Calc 3 13.5

<== Calculus 3
 Number Question Answer See the work 1. Show that the vector field F(x,y,z) = (-8ycos(-x), -xsin(-8y), 0) is not a gradient vector field by computing its curl. How does this show what you intended? curl(F) = &nabla × F = ( , ,). ( 0, 0, -sin(-8y) +8cos(-x)) The work 2. Let F = 1xi + 5yj + 8z k. Compute the divergence and the curl. A. div F = B. curl F = i + j + k A) 14 B) 0i + 0j + 0k The work 3. A) Consider the vector field F(x,y,z) = (-2yz, -6xz, 3xy). Find the divergence and curl of F. . ( , ,) B) Consider the vector field . Find the divergence and curl of F. . ( , ,) A) 0 ( 9x, -5y, -4z) B) -32x - 24y - 6z ( -6(x+y+z), 6(x+y+z), -18(x+y)) The work 4. 1. Let F = 10xi + 3yj + 5zk. Compute the divergence and the curl. A. div F = B. curl F = i + j + k 2. LetF = (4xy, 3y, 6z). The curl of F = ( ,, ) 1. div F = 18 curl F = 0i + 0j + 0k 2. curl of F = ( 0, 0, -4x) The work 5. Let F = (8yz)i + (7xz)j + (6xy)k. Compute the following: A. div F = B. curl F = i + j + k C. div curl F = A) 0 B) -xi + 2yj - zk C) 0 The work 6. Consider the vector field F(x,y,z) = (-7y, -7x, 3z). Show that F is a gradient vector field F = &nabla V by determining the function V which satisfies V(0,0,0) = 0. V(x,y,z) = The work 7. Let . Find a function f so that F = &nabla f, and f(0,0,0) = 0. The work 8. For each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f (that is, &nabla f = F). A. F (x, y) = (-16x + 4y)i + (4x + 6y)j f (x, y) = B. F (x, y) = -8yi - 7xj f (x, y) = C. F ( x, y, z) = -8xi - 7yj + k f (x, y, z) = D. F (x, y) = (-8sin(y))i + ( 8y - 8xcos(y) )j f (x, y) = E. f (x, y, z) = A) B) Not Conservative C) D) E) The work 9. Let F = -4yi + 4xj. Use the tangential vector form of Green's Theorem to compute the circulation integral where C is the positively oriented circle . 628.319 The work 10. Let F = 4xi + 2yj and let n be the outward unit normal vector to the positively oriented circle . Compute the flux integral . 18.85 The work 11. Apply the Laplace operator to the function . . The work