Please only answer ques 5-9

3:27 Reset Done 8. [7 marica] Use Stokes' theorem to compute the surface integral N - / (V xG) . ds for the voctor field G(z, y. 2) = (ever.x3=), where & is the half of the ellipsoid 43" + 13 + 423 = 4 that lies to the right of the ze-plane, oriented in the direction of the positive y-axis. 9, [7 marks] Use Gauss theorem to compute the surface integral P= $ J .as for the vector held J(r. y, =) = [xjx. where x - (r, y, =), and S consists of the hemisphere = = 1 -r- -y and the dink Ity'si 10. (8 marks] Let A be the portion of the elliptic cylinder described by the vector function ra. B = (2cus d.min B.a), DSasa, OSAS2. Calculate the flux 4 of the vector field (r.y, 2) = (6 , -y.Isin=) acrew A in the direction of positive orientation.1. [8 marks] Let up ry and v = ap. Using this change of variables, curaputs the double integral hoki usids dy where Q is the region onclosed by the four curves sy = 1. my -2x, ry= 1 and my= 2. 2 8 marks) Calculate the volume of the solid region I' bounded by the come = = 2\\ F + y and the paraboloid |7 marks Compute the triple integral where It is the solid region in the first octant bounded by the sphere ? by + : - I and the three planes 4. (6 marks] Compare the path integral. where C is the line segment from (0,0.0) to (1, 2,31 5. (akey Use Green's theorem to compute the path integral where T Is the triangle with vertices (0. 1). (1. 1) and (1. 2), oriented counterclockwise. I Consider the force field whose x(z, v) is an arbitrary smooth function, and Fle,vi = (amay + you.zomy - puny). Note that F(m. y) is almost always a non-conservative force field, canopt for certain choices of function y(nip). (1) [6 marks] Determinn a pop-trivial function x(z, ") for which the force field F(r, y) is a conservative force field, for all mal whims of a nad y. (M) (5 marks] Constrict a potential faction ofry) for this conservative force Bold F(r, p), defined using non trivial function x(5, ") determined above (ul) [3 marks] Calculate the work done ,1 by this commureative force field F(s, y) in moving a particle along some amonth curve from (1, 1) to (#, 2m). 7, 15 marks] Compute the surface integral where I in the portion of wie comes = \\/2: + p that In between the two burtantal plane = = 1 and = = 2