For what simple closed (positively oriented) curve C in the plane does the line

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For what simple closed (positively oriented) curve C in the plane does the line integral have the largest positive value? (Use Green's theorem). Determine the largest value. Let C be the unit circle, oriented counterclockwise, and consider the vector field F = (xy , y2). Find the flux of F through C by direct calculation (evaluating a line integral). Find the flux of F through C using Green's theorem. Consider the region R enclosed by the x-axis. x = 1 and y = x3. Use the normal form of Green's theorem to find the flux of F = (0. 1 + y2) out of R. Find the flux out of R through the two sides C1 (the horizontal segment) and C2 (the vertical segment. Use parts (a) and (b) to find the flux out of the third side C3 Evaluate where S is the part of the cone that lies inside the cylinder x2 + y2 = 9 Evaluate where F = (x, y, 0) and S is the upper hemisphere Note: n is the upward pointing normal vector to S. Let F = (exyz, exz + 2yz. exy + y2 + 1). Show that F is conservative. By a systematic method (not guess and check), find a potential tor F. Let S be the part of the spherical surface x2 + y2 + z2 = 4 lying in x2 + y2 > 1. i.e. outside the cylinder of radius one with axis the z-axis. Compute the flux outward through S of the vector field F = (y, -x, z). Show that the flux of this vector field through any part of the cylindrical surface is zero. Using the divergence theorem applied to F, compute the volume of the region between S and the cylinder. Use the divergence theorem to compute the flux of F = (1, 1, 1) outwards across the closed surface x4 + y4 + z4 = 1. Let S be the part of the surface z = xy, where x2 + y2 1. Compute the flux of F = (y, x, z) upward across S by reducing the surface integral to a double integral over the disk x2 + y2 1. Let S be the surface formed by part of the paraboloid z = 1 - x2 - y2 lying above the xy-plane and let F = (x, y, 2(1 - z)). Calculate the flux of F across S, taking the upward direction as the one for which the flux is positive, by: Direct calculation of Computing the flux of F across a simpler surface and using the divergence theorem.

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