advertisement

Name ID MATH 311 Section 200 1. Exam 3 Spring 2001 P. Yasskin 1 /20 3 /30 2 /10 4 /40 20 points Let V be the vector space of functions spanned by v 1 t and v 2 t 2 with the inner product 1 f, g ; ft gt dt 0 a. 10 Find cos 2 where 2 is the angle between v 1 and v 2 . b. 10 Apply the Gram-Schmidt procedure to £v 1 , v 2 ¤ to produce an orthonormal basis £u 1 , u 2 ¤. 1 2. 10 points In R 5 find the volume of the parallepiped with edges a 1, 0, 2, 0, 1 b 0, 2, 1, 0, "1 c "1, 0, 0, 2, 1 2 3. 30 points A paraboloid in R 4 with coordinates w, x, y, z , may be parametrized by r, 2 r cos 2, r sin 2, r 2 , r 2 w, x, y, z R for 0 t r t 3 and 0 t 2 t 2=. a. 10 Find the area of the surface. b. 10 Compute P ;; 1 8w2 8x 2 dS over the surface. c. 10 Compute I ;;xy dw dz " wz dx dy over the surface. w r cos 2, x r sin 2, y r2, z r2 3 4. (40 points) The solid paraboloid V at the right is given by x 2 y 2 t z t 4. It’s boundary (denoted by V) has two parts: The paraboloid P given by z x 2 y 2 for z t 4. The disk D given by x 2 y 2 t 4 with z 4. xz 2 , yz 2 , zx 2 y 2 . Let G G . 5 Compute a. b. G dV over the solid paraboloid V. 10 Compute ;;; V HINT: Use cylindrical coordinates. 4 c. dS over the paraboloid P with normal pointing DOWN and OUT. 15 Compute ;; G Pw HINT: Parametrize the paraboloid with coordinates r, 2 . d. dS over the disk D with normal pointing UP. 5 Compute ;; G Du HINT: Parametrize the disk with coordinates r, 2 . e. dS ;; G dS ;; G dS 5 Compute ;; G V Pw Du (Note: By Gauss’ Theorem, the answers to (b) and (e) should be equal.) 5