1. Use the relationship between the triple scalar product, the determinant, and the volume of a parallelepiped to derive the change of coordinates formula for spherical coordinates. That is to say, show explicitly that a(x, y, z) a( p, $, 0 ) = p2 sin p. 2. Suppose that VX F = 0 on R2 \ {(0, 0) }. For any simple closed piece-wise smooth curve C that does not go through the origin, show that there are only two values possible for F . dr. 3. Suppose f and g are continuously differentiable on a closed bounded region R of the plane whose boundary C consists of closed piece-wise smooth curves C1, . . . , Cn with outward unit normal vector field N. Define % to be the directional derivative of f along N, and likewise for that of g. . Show ds = JJ vzfdA. . Show $ 1 09 ds = sveg + ( Vf . Vg)dA. . Show that if f and g are harmonic (their Laplacians are 0) then ag of 9 ON ds = (fV 2g - gV2 f ) dA = 0