MAT 324 Final, Spring 2009

Mat 324-Mathematics for Engineering Final exam 1. Use Green’s theorem to evaluate the work of the force P=(x°43y)i+ (2x-e”" ) j along thecircle(x-1) +(y-5)=4. 2. The electric field due to a point charge q at the origin is given by the force E= kq 7/ [il ,where F=xi ty jtzk, Fl = length of fF . and kis.a constant. Determine the flux of E out of the sphere x? +y? +7?=a? (20 pts) a) Locate the 4 roots of z*=1-i on the appropriate circles b) Write f(z)=Im (.-32)+ zRe(z?)—5z in the form f(z)=u(x,y)+iv(x, y) c) Ife*=2i, then find z. (15pts) a) If f(z)= aa > 465-2 ds, where Cis |z|=3, then what is the value of f+ i)? b) Draw five staple curves C, and C> in the right half plane Re(z) > 0 between z = 1 +i and z=3 + 2i. Then show that [Fee [. Gq 2 c, 2 c) Let C be the circle |z\=1. Evaluate & and feraz forneZ. Cc z 8 (15pts) : . ap. | : . a) Determine the residue of f(z)=(z-i) sin — at z = i by using the Laurent Z-i series of f(z). b) Determine the order of the zero, z = 0, of the function f (z)=2(e* -1) 2z c) Evaluate las where C is the circle [z| =! Z +2z +2z (1Spts) 6. Evaluate the integral [ 2 =] dx x* 45x? 44 (25pts)