MAT 323 Final, Fall 2005

Vector & Tensor Analysis Final Exam (two hours) ‘\ 1) The derivative of S(x,y) at py (1,2) in the direction of A=7 +f is 2V2 and in the direction of B =-27 is —3. What is the derivative of fin the se 27? (10 points) 2) If the rectangular cartesian coordinates system is rotated upward by 30° about the x, -axis. a new system called ox; x} x3 is obtained. What are the components of A = (I, 1,1) in the new \U coordinates system? a . (10 points) 3) A vector field # is expressed in spherical coordinates system by F'= Fie, + Fé, + Fe. ie, a- Write the formula expressing divF. b- Find an explicit formula for the function f(p) if F = f(p¥, and divF =0. (10 points) 4) Let S denote the portion of the paraboloid z = x* + y’ located between the planes z = 0 and . \o z= 4. Find the area of S. (15 vain aN 5) Evaluate the work of F = (e %~ ze" Fa le “F xe? "jj +(e" - ye") along the path C b parameterized by - 1 l-e r(t)= omae > sine, Ine \ (15 points) 6) Let F= (y? +2V + (x? + 2) + (x? + ye and let C denote the boundary of the triangle cut ‘> from the plane x + y+z=1 by the 1* octant. o (15 points) If C is oriented counterclockwise, what is the circulation of F around C? 7) Let F= xy y jt zk, , (x? +t yt zy? (x? tye 2yi (x? tyt 2) \? Find the flux of F across the boundary of the region D described by (x, },z)¢ D iff (a sxtyee sb. Ou f= ix b _—— QO (15 points) = AV2 — 2Ve —n > Ya 7 \" \ (0b) — Vex Dy 8) IfA is a 2™ order antisymmetrical tensor with components a, and the vector b is given by yk ik b, =1& a i=l, 2,3. 2 Show that a,, = &. soi and compute a,,, @,,, 4; in terms of 6,, b,, 6;. . ry (10 points)