Calculus 3 (Math-UA-123) Fall 2016 Homework 10 Due: Thursday, December 8 at the start of class Please give complete, well-written solutions to the following exercises. 1. For each of the following vector fields, determine if the divergence is positive, zero, or negative at the indicated point. Explain/justify your answer. (b) (c) 6 6 6 4 4 4 2 2 0 0 y y (a) -2 2 0 -2 -2 -4 -4 -6 -6 -4 -6 -4 -2 0 2 4 6 -6 -4 x -2 0 2 4 6 x -6 -6 -4 -2 0 2 4 2. (a) Sketch a the vector field F = yi + xj + 0k in the xy-plane. (b) Based on your sketch, what is the direction of rotation of a thin twig placed at the origin along the x-axis? (c) Based on your sketch, what is the direction of rotation of a thin twig placed at the origin along the y-axis? (d) Compute curl F. 3. Prove each identity below, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F and G are vector fields, then F G, and F G are defined by (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (a) div (F G) = G curl (F) F curl (G) (b) curl (curl (F)) = grad (div (F)) 2 F 4. Find a parametrization of the portion of the plane y + 2z = 2 inside the cylinder x2 + y 2 = 1. Use the parametrization to formulate the area of the surface as a double integral. Then, evaluate the integral. p 5. Find a parametrization of the portion of the cone z = x2 + y 2 /3 between the planes z = 1 and z = 4/3. Use the parametrization to formulate the area of the surface as a double integral. Then, evaluate the integral. [Problems 6 and 7 are on the next page.] 1 6 Calculus 3 (Math-UA-123) Fall 2016 6. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz plane about the z-axis. Suppose that C has a radius r and center (R, 0, 0). (a) Find a parametrization r(u, v) of the torus. Specify the set D in which (u, v) must lie. Hint: You can choose let u represent the angle that the line from the point r(u, v) on the torus to the center of the rotated circle form with the xy-plane, and let v denote the angle formed by the line from the point r(u, v) on the torus to the origin with the positive x-axis. See figures below. (b) Show that the surface area of the torus is 4 2 Rr. p 7. Integrate g(x, y, z) = x y 2 + 4 over the surface S that is the portion of the surface y 2 + 4z = 16 that lies between the planes x = 0, x = 1, and z = 0. 2