Problem 2. /question purpose: to build familiarity with parametrizations and to extend un- derstanding of major theorems of vector calculus) Consider the same vector field F as in the previous problem. Let S2 be the surface z = 9 - (x2 + y?) with = 2 0. (a) Sketch S2 and the curve C from the previous question. b) Find a parametrization of the surface S, in terms of two variables u and v. (c) Find or x ar (d) Write down the surface integral of the vector field V X F on S2, in terms of your parametrization. /do not attempt to evaluate the integral). You found V x F in the previous problem. Question 1, and of x . in Question 2.c (e) On your sketch of $2, show the direction of the unit normal for the integral that arises from your parametrization (used in the integral) you just need to mark the vector at a few points, see page 9 of the Bundoora lecture 17 notes for an example). (f) Using whatever method you would like, find the value of the integral from (d). It may help to use your results from the previous question and one of the important theorems of vector calculus. Justify your answer. Problem 3. /question purpose: extend understanding of major theorems of vector calculus) A pipe with radius R is aligned with the z direction. It is filled with fluid flowing in the positive z direction. The velocity of fluid in the pipe is given by u(r, 0, z) which varies with location. Consider the cylindrical region given by p - R, 0