Math_317_April_2005

Math_317_April_2005 - April 2005 MATH 317 Name Page 2 of 9...

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Unformatted text preview: April, 2005 MATH 317 Name Page 2 of 9 pages Marks [12] 1. Let C be the ()seulating circle to the helix r : (cos t, sin 15,15) at the point Where : 7r/ Find: 15 (a) the radius of curvature of C; (b) the center of C; ( e) the unit normal vector to the plane of C. Continued on page 3 April, 2005 MATH 317 Name Page 3 of 9 pages [13] 2. Let F : (yze()s:1;,zsin:1; + 2yz,y sin :1; + 3/2 i sin 2) and let C be the line segment r(t) : (t,t,t), for 0 g t g 7r/2. Evaluate / F 0 dr. . (7 Continued on page 4 April, 2005 MATH 317 Name Page 4 of 9 pages [13] 3. Let S be the ellipsoid 1‘2 + 23/2 + 322 : 16, and n its outer unit normal. " I). .4 7 2.1.1 (21) Find F o ndS if F(:1;,y,2) : [(l’yf) ’ ’ ) I 3/2. ~s K$imz+wglV+wzihd ('7’): Z) i 2: Klim2+wgav+wzimaw” (b) Find GondS if G(:1;,y,z) : . . S Continued on page 5 April, 2005 MATH 317 Name Page 5 of 9 pages [13] 4. Find the net flux F o ndS of the vector field F(:1;,y, 2) : (:1;,y, 2) upwards (with . . S 2 2 respect to the z—axis) through the surface S pararneterized by r : (m; ,71, 71,7141), for 0371,31,0§7}§3. Continued on page 6 April, 2005 MATH 317 Name Page 6 of 9 pages [13] 5. Let F : (sin $2,112, 22). Evaluate % F 0 dr around the curve C of intersection of the . (7 cylinder 1‘2 + 3/2 : 4 with the surface 2 i 1‘2, traversed counter clockwise as Viewed from high on the 2—axis. Continued on page 7 April, 2005 MATH 317 Name Page 7 of 9 pages [13] 6. Explain how one deduees the differential form 1‘ VXE:7—d{—H cdt of Faraday’s law from its integral form . 1d .. yiEodr——//HondS. .(2 Cdt..s Continued on page 8 April, 2005 MATH 317 Name Page 8 of 9 pages [13] 7. Let Q C 123 be a srnoothly bounded domain, with boundary [)9 and outer unit normal n. Prove that for any vector field F which is continuously differentiable in Q U 09, // VXFdV// FXndS. ...o H852 Hint: Recall the fundamental lernrna used in proving the divergence theorern. Continued on page 9 l10l April, 2005 8. MATH 317 Name Page 9 of 9 pages True or False. Explanations are not required. Consider vector fields F and scalar functions f and g which are defined and smooth in all of three—dimensional space. Let r : (11;, y, 2) represent a variable point in space, and let u: : (M1, M2,w3) be a constant vector. Let Q be a srnoothly bounded domain with outer unit normal n. Which of the following are identities, always valid under these assumptions? (a) Von:0 (b)F><Vf:fVoF (C) sz:V(V°f) (d) VXVf:0 (0) (VXf)+(VX9):Vvag (f) VOVXF:0 i, : 0, for r yé 0 M2 (g) V- (h) V><(w><r):0 (1) fVoFdV: VfoFdVJr/éan . ndS <J> [/99 W11}; W The End Be sure that this examination has 9 pages including this cover The University of British Columbia Final Examinations — April, 2005 Mathematics 31 7 Section 201 J. Heywood Closed book examination Time: 2.5 hours Name Signature Student Number— Instructor’s Name Section Number Special Instructions: Calculators and books are not permitted. One 8%” X 11” two—sided page of notes is permitted. Rules governing examinations 1. Each candidate should be prepared to produce his library/AMS card upon request. 2. Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expir ration of one half hour, or to leave during the first half hour of the examination. 13 Candidates are not permitted to ask questions of the invigilators, except in COM 13 cases of supposed errors or ambiguities in examination questions. CAUTION 7 Candidates guilty of any of the following or similar practices A; 13 shall be immediately dismissed from the examination and shall be liable to . . . . l" ‘ disciplinary action. 0 (a) l\'laking use of any books, papers or memoranda, other than those aur thorized by the examiners. V . . . . . 5 (b) Speaking or communicating With other candidates. 7 (c) Purposely exposing written papers to the View of other candidates. The plea of accident or forgetfulness shall not be received. 3. Smoking is not permitted during examinations. Total ...
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Math_317_April_2005 - April 2005 MATH 317 Name Page 2 of 9...

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