Calculus of Several Variables (TEN A), (TEN 1)

Transcription

1 Famil name: First name: I number: KTH Campus Haninge EXAMINATION Jan 6 Time: Calculus o Several Variables TEN A TEN Course: Transorm Methods and Calculus o Several Variables 6H79 Ten Ten A Lecturer : Armin Halilovic INSTRUCTIONS: Write our name and I-number on each page. Allowed to use : Calculator You are NOT allowed to use tables o mathematical ormulas. Use o an communication device is strictl prohibited when taking this eamination. Grading: For each correct solution 4 points will be awarded. Credit will be given or presentation and methods o solutions. A maimum o 4 points can be earned. Points can also be deducted or unsubstantiated answers. You can get ma 4 points. At least points are required to pass the eam. This piece o paper our questions ou hand in together with our solutions. Grading scale : Total points: 4-7 points are required or the grade ; 8 - points are required or the grade 4; -4 points are required or the grade 5 There will be a supplementar eam or those students who can manage at least 9 points in the eam. The supplementar eam will be dierent rom the regularl scheduled eam though it will cover the same material Questions:. Which one o the ollowing unctions a 5 4 b sin c ln satisies the Laplace equation?. The temperature T at points o -plane is given b T.

2 a raw a contour diagram or T showing some at least three isotherms curves o constant temperature b In what direction should an ant at position move i it wishes to cool as quickl as possible? c I the ant moves in that direction at speed k units distance per units time at what rate does it eperience the decrease o temperature?. etermine which one o the epressions below is well-deined and calculate it a div rot grad F b rot grad div F and c div grad rot F i F a Calculate the volume o the solid ling between over the domain 4 and b Calculate the volume o the solid K { : e } 5. a For the vector ield F ind the potential U which satisiesu 4. b Let F. Evaluate the line integral Fdr along the quarter o the circle o unit C radius centered at the origin and traversed counterclockwise. 6. a Calculate the total lu bottom top clindrical wall o the vector ield F 5 i 5j 5k outward through the surace o the clinder a. b Calculate the lu o the vector ield F outward through the b bottom b top b clindrical wall o the clinder. Good Luck!

3 Solutions: Questions:. Which one o the ollowing unctions a 5 a b sin 4 b c ln c satisies the Laplace equation? a 4 Answer a The unction a does not satis the Laplace equation. b sin 6 Answer b The unction b does not satis the Laplace equation. c. Answer: Onl the unction ln c satisies the Laplace equation.. The temperature T at points o -plane is given b T. a raw a contour diagram or T showing some at least three isotherms curves o constant temperature b In what direction should an ant at position move i it wishes to cool as quickl as possible? c I the ant moves in that direction at speed k units distance per units time at what rate does it eperience the decrease o temperature? a 4... k k

4 b gradt i 4j gradt 4 i 4 j An ant at should move in the direction o gradt that is in the direction 4i 4 j in order to cool o as rapidl as possible. c The rate o change in the direction o the vector v 4i 4 j v 4i 4 j 8 v gradt 4i 4 j 4 degrees per unit distance. v 4. etermine which one o the epressions below is well-deined and calculate it a div rot grad F b rot grad div F and c div grad rot F i F 5. To answer the question we use the ollowing properties or the operators grad div and rot curl : grad scalar unction vector unction div vector unction scalar unction rot vector unction vector unction a The epression a is not well-deined because the operator grad is not deined on the vector ield. b The epression b is well-deined. c In the epression c we can calculate rotf but the result is a vector ield an thereore we can not calculate grad rot F. Hence the epression c is not well-deined. So onl the epression b is well-deined and we calculate it: divf 4. grad div F and rot grad div F.

5 4. a Calculate the volume o the solid ling between over the domain 4 and b Calculate the volume o the solid K { : e } a V dd 9 The domain is deined b 4 dd and we change to polar coordinates Jr. π π 9 θ 9 θ r V d r rdr d 9r r dr π b Solution b: V dd e dd The domain is deined b : { } r 67π so we change to polar coordinates Jr. π / π / π / r d ϕ e rdr r d ϕ [ e ] 67π π Answer: a V b V e e π e dϕ

6 5. a For the vector ield F satisiesu 4. b Let F. Evaluate the line integral C radius centered at the origin and traversed counterclockwise. a To ind U we solve the sstem: U U U eq eq eq eq implies U d g * To get g vi substitute * in eq ind the potential U which Fdr along the quarter o the circle o unit U g ý g ý g h. This and * implies that U h ** To ind g vi substitute ** in eq U h h C This and ** implies U C. Thus F is a conservative vector ield and U C is a potential unction or the ield. Since U 4 we have C and U Answer a U b The curve can be parametried as cos t sin t r t cost sin t F t sin t cost and r t sin t cost Hence F t r t sin t cos t π / π / Fdr F t r t dt dt π C Answer b Fdr C π π t so that

7 6. a Calculate the total lu bottom top clindrical wall o the vector ield F 5 i 5j 5k outward through the surace o the clinder a. b Calculate the lu o the vector ield F outward through the b bottom b top b clindrical wall o the clinder. N N N a Total lu outward We use the ivergence Theorem to calculate the total lu outward through the surace o the clinder : P Q R Since div F 5 we have b the ivergence Theorem Φ div F ddd 5ddd 5 Volume K 5 6a π 9a π total b The bottom S K Since outward at the bottom means downward we have N bottom F 55 5 F N 5 5

8 Φ 5dd 5Area 5a π bottom b The top N top F 555 F N 5 Φ top 5dd 5Area 5a π b The clindrical wall Since Φ Φ Φ Φ we have total top botom wall Φ Φ Φ Φ 6a π. wall total top botom