Problem 1 The vector surface integral on the unit sphere Let $3 denote the unit sphere, $3 = {(x, y, 2) : x2 +y? + 23 =1}, with orientation given by the outward normal vector, and let F : S' -> R* denote a vector field defined on $2. Parametrization spherical coordinates, +(0, $), and show that the normal vector To x To = sino (0, $); that is, the normal vector at (8, () points in the same direction as the position vector d(0, $) (and equals the position vector d(0, $) multiplied by sin $). This is a special property of the sphere (draw it to convince yourself). Let F.(0, $) = F(D(0,$)) . (0, $) denote the radial component of F at the point @(0, $) (since $ is the unit vector pointing in the radial direction, the dot product of F and d gives the radial component of F). Show that 1. F.as = [ . F. (0, $) sin odode. Remark. This problem shows that when computing the surface integral of a vector field out of a sphere, only the radial component of the vector field matters, since the normal vector field to a sphere is radial. As an example, we used this in Lecture 16 to compute the electric field due to a point charge