Let S be the surface parametrized by Phi(u, v) = (u cos v, u sin v, u2 + v2). Find a vector normal to S at the the point Phi (1, 0) on the surface. Find an equation for the plane tangent to S at the the point Phi (1, 0) on the surface. Find an equation for the plane tangent to the surface Phi(u, v) = (uv, u2v, v2) at the point P = (2, 2 ,4) on the surface. Evaluate the surface D (x + y) dS where D is the part of the plane x + 2y + 3z = 6 above the triangle in the xy-plane with vertices at (0, 0, 0), (1, 0, 0), and (1, 1, 0). Determine the area of the part of the paraboloid f(x, y) = a2 - x2 - y2 (a is a positive constant) above the xy-plane. Hint: Use polar coordinates. Evaluate the surface H y2 z dS where H is the surface given parametrically by Phi (u, v) = (u cos v, u sin v, u), 0 le u le 1, 0 le v le pi. (Hint: Some of the grunge work for this problem is done in example 4, page 964, of the text.)
