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- Let W be the region between the sphere of radius 4 and the cube of side 1, both centered at the origin. What is the flux through the boundary S = aw of a vector field F whose divergence has the constant value div(F) = -4?Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.Let a vector field be given. What is the integral representing the work done by F along a curve C which is the parabola , from (0,0) to (1,1) ?
- Give an example of a vector field F (x, y, z) that has value 0 on a surface and such that curl F is nonzero everywhere. Be sure to identify the surface and compute the curl.Verify Green’s Theorem for the vector field F(x, y) = (x^2*y^2 , xy) and the closed curve C consisting of the arc of parabola y = x^2 from (0,0) to (1,1) and two line segments from (1,1) to (0,1) and from (0,1) to (0,0).Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl F is nonzero everywhere. Be sure to identify the point and compute the curl.
- Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z) such that curl(G) = F. Such a field G is called a vector potential.Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work done of the vector field F=〈2xy,x2+2〉over the two curves found in part (a).A vector field is given by F = ((1 + xy)exy, x2exy). Compute the integral of the vector field over the curve r(t) = (cos(t), 2sin(t)), where t ranges from zero to π/2.