1. Determine a parametrization for each of the following curves: the intersection of (b) the planes 4x - 3y + 5= = 1. a + y - 2z = 6 (g) the plane 4x - y + = = 3 and the circular cylinder (j) surfaces a - y' + = = 0 and a - y + 2= = 0, between (6, 2, -2) and (1, 1,0)1. In each case evaluate the surface area A of the surface S using (5). (a) HINT: Use = cosv, y = sin v, z = 1, (b) Let S be the circular cylinders' ty* = 1, between z = and 2 = 1 - #'. HINT: Same hint as in part (a).2. Show by triple integration that the volume of the right cir- cular cone shown in Fig. 5 is V = (ah*/3) tana: (a) using cylindrical coordinates (b) using spherical coordinates.duy our curl v = k . Ox (4) Dy Consider, heuristically, the motion of the little rectangular element of fluid shown in Fig. I. At time & it is in position 1. If the element were rigid, its motion would consist of a translation plus a rotation. If so, then at time & + At it might do be in configuration 2. say, due to a translation OO' plus a rotation do. and its angular velocity (taken as positive counterclockwise. with the right-hand rule used to assign a vector direction) would be w ~ (Aa/At)k. However. a fluid element is deformable, not rigid, so it may also suffer a shear deformation as indicated in the Ax configuration 3. What. then, is the fluid's "angular velocity" to mean? It is defined, in fluid mechanics. as the average angular velocity of initially perpendicular edges such as O A and OB. Then Figure I. Plane motion of fluid element. angular velocity of OA = U.(A) - vy (0) Ar k - duy k as Ax - 0, (5a) angular velocity of OB - Us(0) - U.(B) k - - dy duck as Ay - 0, (5b) Ay so that the average of the two = w = 101 1 Dy k (6) In case (5a) is not clear. consider Fig. 2. The average of the a velocity com- ponents or(O) and a.( A) produce the -wise translation of OA, their difference produces the stretching or contracting of OA. and the average of vy (O) and vy(A) produce the y-wise translation of OA. But the angular velocity of O A is due to the difference of *(O) and u( A). It is the vertical velocity of A relative to O. namely, vy(A) - vy(O) divided by the radius OA of that motion. Similarly, the angular "Y ( B ) velocity of OB is v.(O) - ur( B) divided by OB. Comparing (4) and (6). we see that curly = 2w. from which we draw the B ".(B) following simple and important physical interpretation of the curl of a vector field: curl v( P) is twice the angular velocity of the fluid at P. ",(A) Although our (heuristic) proof of the italicized claim was for plane flows, it is not hard to show that that result holds for nonplanar flows as well. And if our ",(O) A vector field is not a fluid velocity field (e.g.. it might be an electric field, magnetic field. or gravitational force field) we can at least think of it as a velocity field in Figure 2. Angular velocities order to have access to the physical interpretation stated above in italics. of O A and OB