Engineering Mathematics (E35 317) Exam 3 November 7, 2007

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1 Engineering Mathematics (E35 317) Exam 3 November 7, 2007 This exam contains four multile-choice roblems worth two oints each, twelve true-false roblems worth one oint each, and four free-resonse roblems worth 20 oints altogether, for an exam total of 40 oints. Part I. Multile-Choice Clearly circle the only correct resonse. Each is worth two oints. # # 1. Let 0ÐBß Cß DÑ œ BC D, and observe that f0ðbß Cß DÑ œ ÒC ß #BCß "Ó. Find the value of the following line integral. (A)! (B) " (C) #" (D) #& (E) &% (F) &( () (# (H) (% (I) *$ (J) *& Ð%ß&ß'Ñ ' # Ð"ß#ß$Ñ ÐC.B #BC.C.DÑ. 2. The vector equation below is a arameterization of which of the following surfaces? r Ð?ß@ÑœÒ@ cos?ß@ sin?ß@ó!ÿ?ÿ# 1!Ÿ@Ÿ% (A) circular cylinder (B) disk (C) cone (D) shere (E) lane (F) none of the above

2 3. Consider the following differential equation. # ww " w B B # % Ð" B ÑC C C œ! Let B! œ!. According to the alicable convergence theorem, any series in the solution is guaranteed to converge at least on what interval? (A) Ð!ß "Ñ (B) Ð "ß "Ñ (C) Ð!ß #Ñ (D) Ð #ß #Ñ (E) Ð!ß %Ñ (F) Ð %ß %Ñ () Ð!ß Ñ (H) Ð ß Ñ 4. Consider the following differential equation. ww w BC ÐB #ÑC ÐB $ÑCœ! What must the form be for a basis of solutions C" and C# about the oint B! œ!? # 7 " 7 (A) C œ B! + B C œ B! E B # 7 " 7 (B) C œ B! + B C œ 5C ln B B! E B # 7 7 (C) C œ B! + B C œ! E B # 7 7 (D) C œ B! + B C œ 5C ln B! E B $ 7 " 7 (E) C œ B! + B C œ B! E B $ 7 " 7 (F) C œ B! + B C œ 5C ln B B! E B $ 7 7 () C œ B! + B C œ! E B $ 7 7 (H) C œ B! + B C œ 5C ln B! E B

3 Part II. True-False Write out the word true or false for each of the following. Each is worth one oint. 5. Consider the motion of an incomressible fluid in a region in three-sace. If the region contains no sources or sinks, then div u œ! throughout the region (where the function u is the roduct of the density and the velocity functions). 6. The velocity vector field given by v ÐBßCßDÑœÒC $ D # ß$BC # ß#BDÓ is irrotational. 7. If a curve is arameterized in two different ways (with the same orientation), then the value of a line integral over will not change. (Note that this is not asking about two different aths, but about two different arameterizations of the same ath.) 8. The following differential form is exact. B B B B Ð/ sin C / cos DÑ.B Ð/ cos CÑ.C Ð/ sin DÑ.D For roblems 9 through 11, refer to the following region V in the BC-lane, which is bounded by # # the circle B C œ ", the line C œ B, and the B-axis.

4 9. The integral '' 0ÐBßCÑ.E can be comuted correctly as follows. V! È" B " È" B È# # B!! # # ' ' 0ÐBß CÑ.C.B ' ' 0ÐBß CÑ.C.B 10. The integral '' 0ÐBßCÑ.E can be comuted correctly as follows. V " È " C# ' '! C 0ÐBß CÑ.B.C 11. The integral '' 0ÐBßCÑ.E can be comuted correctly as follows. V ' $ 1 % " '!! 0Ð< cos ) ß< sin ) Ñ.<. ) # # 1 1!! ' ' % ' ' ' ' % cos B sin C.C.B œ sin C.C cos B.B 13. When 8 is a nonnegative integer, Legendre's equation of order 8 has a basis of solutions where both solutions are olynomials. 14. When 8 is a ositive odd integer, the Legendre olynomial T8ÐBÑ is an odd function, and when 8 is a ositive even integer, the Legendre olynomial T ÐBÑ is an even function >Ð!Ñ œ "x 16. > Ð"!!Ñ œ ** > Ð**Ñ

5 Part III. Free Resonse In each roblem in this section, follow directions carefully. The oint value for each roblem is shown to its left. (2) 17. Suose F is a continuous vector function with continuous artial derivatives in a simly connected domain H in $. (All these details are needed simly to guarantee that everything is nice. ) We discussed four theorems, each of which gave a condition for the ath indeendence of the line integral ' F. r. One of them was, The line integral ' F. r is ath indeendent if and only if F. r is exact. List two others. (You may recall that the last of the four theorems required an extra condition, but don't worry about that. Under the above assumtions, that theorem, like the others, can be stated as if and only if. ) The line integral ' F. r is ath indeendent if and only if The line integral ' F. r is ath indeendent if and only if (2) 18. Let F œò sin # Bß/ CD ßC # ln BÓ. Let W be the surface of the cube "ŸBŸ#ß #ŸCŸ$ß $ŸDŸ%. Set u the integral or integrals which would enable you to find the value of the surface integral '' Ð F n Ñ.E. Do the comlete set-u so that the very next ste would be W integration, but do not do the integration or finish the roblem. (Do not worry about the issue of orientation.)

6 (7) 19. Let F œ ÒBCDß BCß BÓ. Let be the curve in three-sace which goes from Ð"ß "ß $Ñ to Ð#ß "ß &Ñ to Ð#ß $ß (Ñ to Ð"ß $ß &Ñ and back to Ð"ß "ß $Ñ, all in straight line segments. Hint: This traces out a arallelogram in the lane Dœ#B C. Set u the integral or integrals which would enable you to find the value of the line integral F. r. Do the comlete set-u so that the very next ste would be integration, but do ) not do the integration or finish the roblem. (Do not worry about the issue of orientation.)

7 (9) 20. Find two linearly indeendent ower series solutions of the following differential equation about the regular oint B œ!. ww w C %C BC œ!! As you work through the rocess, find the coefficients u to and including + %. Show all your work. Make sure your final answer consists of two secific series C" and C# with no arbitrary constants. (1) Extra Credit What is my favorite color?