
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Let F be the vector field on R3 given by F(x,y,z)=(2xz,-x,y^2)
evalute the volume integral below. cheers
19. Let F be the vector field on R given by F(r,y,z) = (2xz, -x, y2) Evaluate 2xzdV, FdV xdV where V is the region bounded by the surfaces 0, y = 6, z = x2 and z = 4. 0, y
5. (7 points) Let f: R3 → R be the function f(x,y,z) = x2 + y2 +3(2-1)2 Let EC R3 be the closed half-ball E = {(x, y, z) e R$: x² + y2 +< 9 and 2 >0}. Find all the points (x, y, z) at which f attains its global maximum and minimum on E.
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
Consider the vector field F(x, y, z) = 8x^2 + 3y, −5x^2y − 4y^2, 6x^2 + 7y − 8 which is defined on all of double-struck R3, and let F be the rectangular solid region F = {(x, y, z) | 0 ≤ x ≤ a, 0 ≤ y ≤ b, −1 ≤ z ≤ 1} where a > 0 and b > 0 are constants. Determine the values of a and b that will make the flux of F...
*1. Let S2((x, y, z) e R3:xy+2 be the unit sphere and let A: S2 S, be the (antipodal) map A(x, y, z)-(-x,-y,-z). Prove that A is a diffeomorphism.
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0, 2 0} S = = Identify a parametrization d: U -> S of S (so UC R2 open so that S is part of a cone. etc.) such that d 1 is a conformal chart Suggestion: parametrize as a surface of revolution.
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0,...
Problem 2 [10pts] Let f : R3 + R2 be a linear transformation given by f((x, y, z) = (–2x + 2y +z, -x +2y). Find the matrix that corresponds to f with respect to the canonical bases of R3 and R2.
Let f: R3 → R3 be the matrix transformation defined by Findx,y,andzsothat,(2):[3] (Usetheparametertasnecessary.) 10 17 (Use the parameter t as necessary.) (x, y, z) = (1 2