3c and 3d only.

3) Let M be the portion of the upper-half of the sphere x2 + y + z = 2 outside the cylinder x2 + y = 1 in the region with y 2 0. In other words, M = ( (x, y, z) : x2 + y + z? = 2, x2 + y> > 1, y 2 0, z 20). (a) Use spherical coordinates to find a parametrization (, 0) for M. Write down the normal field N = D X De associated with your parametrization and indicate whether this normal field is outward or inward. (b) Use your parametrization in (a) to find the area of M. (c) Consider a fluid with density lyl M(x, y, Z ) = - x2 + 2+ 22 measured in g/in' flowing with velocity v( x, y, Z) = (0, x, 1) measured in in/s. Compute the amount of fluid that passes through our surface M going towards the origin after 3 seconds. In other words, compute the surface integral 3 HU . Nas , where N is a vector field consisting of inward normal vectors to M. (d) Note that the quantity in (c) is a negative number. What is the physical meaning of this negative sign