Code No. : Sub. Code : GMMA 21/ GMMC 21

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1 Reg. No. :... ode No. : 08 Sub. ode : GMMA 1/ GMM 1 B.Sc. (BS) DEGREE EXAMINATION, NOVEMBER 016. Second Semester Mathematics Main VETOR ALULUS (Also common to Maths with omputer Applications) (For those who joined in July ) Time : Three hours Maximum : 75 marks PART A (10 1 = 10 marks) Answer ALL questions. hoose the correct answer. 1. f u iˆ uj ˆ 5 u kˆ df GÛÀ ß v du (A) (B) () (D) uiˆ 10 u kˆ uiˆ 5 u kˆ uiˆ u 10 u kˆ uiˆ 10 u kˆ

2 w If f u iˆ u 5 u kˆ df, then is du (a) (c) uiˆ 10 u kˆ (b) uiˆ 5 u kˆ uiˆ u 10 u kˆ (d) uiˆ 10 u kˆ. vø\ ÁøPUöPÊ Ea\ AÍÁõP UP, Ax Ø μ ö\[szx «x HØ kzx Põn (A) 0 (B) () (D) øápò õä Àø» The directional derivative is maximum when the angle made by it with the normal to the surface is (a) 0 (b) (c) (d) None of these. r xi yj zk GÛÀ, r n r ß v (A) 0 (B) 1 () (D) Page ode No. : 08

3 If r xi yj zk, then r n r is (a) 0 (b) 1 (c) (d). x y z GÛÀ ß v (A) (B) () (D) xy z 6x yz 1x y z y z 6x yz x y z y z 6x yz 1x y z y z x yz 1x y z If x y z, then is (a) (b) (c) (d) xy z 6x yz 1x y z y z 6x yz x y z y z 6x yz 1x y z y z x yz 1x y z Page ode No. : 08

4 w 5. x y, z 0 z, GßÓ E øí ß ÁøÍ Ø μ ß «x EÒÍ Jº A»S ö\[szx öáuhº (A) xiˆ yˆj (B) xiˆ yˆj () xi yj (D) øápò õä Àø» The unit vector normal to curved surface of the cylinder x y, z 0 and z is (a) xiˆ yˆ xiˆ yˆj j (b) (c) xiˆ y (d) None of these 6. f GÝ öáuhº PÍ, õxpõ õú (conservative) PÍ õp UP (A) A GßÓ öáuhº ÒÎa \õº, f A GßÓ Pmk õmkhß UP Ásk (B) GßÓ vø\ ØÓ ÒÎa \õº, f GßÓ Pmk õmkhß UP Ásk () A GßÓ öáuhº ÒÎa \õº, f A GßÓU Pmk õmkhß UP Ásk (D) øápò GøÁ² Àø» Page ode No. : 08

5 A vector field f is said to be a conservative field if (a) (b) (c) (d) There exists a vector point function A such that f A There exists a scalar point function such that f There exists a vector point function A such f A that None of these 7. V Gß x, S GßÓ Ø μ ø Eøh μ hzvß PÚ AÍÄ GÛÀ, r ds ß v (A) V (B) V S () V (D) V If V is the volume of the region enclosed by the surface S, then r ds is S (a) V (b) V (c) V (d) V Page 5 ode No. : 08

6 w 8. x y z a GßÓ PõÍzvÀ À AøμU PõÍzvß PÚ AÍÄ (A) () 1 a (B) a (D) a a The volume of the upper hemisphere x y z a is (a) (c) 1 a (b) a (d) a a 9. Ámh I Áμ õp öpõsh, x y z 1 GßÓ AøμU PõÍzvß «x EÒÍ A yiˆ zj ˆ xkˆ GßÓ öáuh US A. dr ß v (A) 0 (B) () (D) Page 6 ode No. : 08

7 For A yiˆ zj ˆ xkˆ, in the upper half of the sphere x y z 1 and for the boundary circle, the value of A. dr is (a) 0 (b) (c) (d) 10. x a cos, y b sin GßÓ }Ò Ámhzvß μ ÍÄ (A) () a (B) ab b (D) a b The area of the ellipse x a cos, y b sin is (a) (c) a (b) b ab (d) a b PART B (5 5 = 5 marks) Answer ALL questions choosing either (a) or (b). Answer should not exceed 50 words. 11. (A) = xy z iˆ 6 x z xz ykˆ GÛÀ ß v ø U PõsP. Find if xz y kˆ. = 6xy z i x z j Page 7 ˆ ode No. : 08 ˆ

8 w (B) y z 6 1 1, 1 GßÓ ÒÎ À öuõkáøμz uízvß \ ß õmøhu Psk iupä. x GßÓ Ø μ À, Find the equation of the tangent plane to the surface x y z 6 at the point 1 1, (A) A ØÖ B Gß Ú öáuhº ÒÎa \õº PÒ. A B A B B GßÓõÀ A GÚ {ÖÄP. Show that. A B A B B A where A and B are vector point functions. (B) a Gß x J õô¼ öáuhº GÛÀ a r r a GÚ Põs UPÄ. r If a is a constant vector, show that a r r a. r 1. (A) f y y z i x z xy j ˆ y z xkˆ GßÓ öáuhº uí õxpõ õúx GÚ {ÖÄP ØÖ AuÝøh vø\ ¼ ö õmhõß] ø»u PõsP. ˆ Page 8 ode No. : 08

9 Show that the vector field f, where f y y z iˆ x z xy y z xkˆ is conservative and find its scalar potential. (B) x y 6z 6 GßÓ uízvß uà AøμUPõÀ ÁÍõPzvÀ EÒÍ Sv ß Ø μ S ß μ ÍøÁU PõsP. Find the area of the surface S of the portion of the plane x y 6z 6 contained in the first octant. 1. (A) A x z iˆ y z ˆ j z kˆ ØÖ S Gß x x 0, y 0, z 0, x y z 8 Gß ÁØøÓ öpõskòí Ø μ GÛÀ A. ds, ß S v ø U Psk iupä. Evaluate A. ds, where A x z iˆ y z ˆ j z kˆ S and S is the surface bounded by x 0, y 0, z 0, x y z 8. Page 9 ode No. : 08

10 w (B) x y z 1 GßÓ PõÍzvß Ø μ S GÛÀ, Põì øháºásì uøózøu ß kzv S zdy dz y zdz dx x y dx dy x ß v ø U PõsP. Apply Gauss divergence theorem to evaluate S zdy dz y zdz dx x y dx dy x where S is the surface of the sphere x y z (A) Gß x, x y 1, z 1 GßÓ Ámh GÛÀ, ì hõu uøózøu ß kzv ( 1 y) z iˆ 1 zx 1 xy kˆ d r ß v ø U Psk iupä. By using Stoke s theorem evaluate the Integral ( 1 y) z iˆ 1 zx 1 xy kˆ d r where is the circle x y 1, z 1. Page 10 ode No. : 08

11 (B) Gß x x 0, y 0, x y 1, y 0 GßÓ PõkPøÍU öpõskòí U Põn μ ß Áμ GÛÀ x 8y dx y 6xy v ø U PõsP. dy ----ß Evaluate x 8y dx y 6xy dy, where is the boundary of the triangular area enclosed by the lines x 0, y 0 x y 1. PART (5 8 = 0 marks) Answer ALL questions choosing either (a) or (b). Answer should not exceed 600 words. 16. (A) P GßÓ ÒÎ À ß vø\, P GßÓ ÒÎ «x EÒÍ GßÓ \ uí Ø μ ØS ö\[szuõp US GÚ {¹ UPÄ. ¾ P GßÓ ÒÎ À, ß õn, P GßÓ ÒÎ À EÒÍ vø\ ÁøPUöPÊÂß Ea\ AÍÄ GÚ {¹ UPÄ. Page 11 ode No. : 08

12 w Prove that the direct of at P is normal to the level surface through P. Also prove that the magnitude of at P is the maximum of the directional derivative of at P. (B) (i) f u GßÓ öáuhº ÒÎa\õº õóõu vø\ öpõsk UP, ßÔ ø õu df ØÖ õx õú Pmk õk f 0 du (ii) GßÖ Põs UPÄ. f u GßÓ öáuhº ÒÎa \õº õóõu vø\ø U öpõsk df d UPÂÀø»ö ßÓõÀ, f du du GÚU Põs UPÄ. (i) Show that the necessary and sufficient condition for the vector point function f u to have constant direction df if f 0. du (ii) Show that, if f is not of constant df d direction, then f. du du Page 1 ode No. : 08

13 17. (A) (i) A Gß x J öáuhº ÒÎa \õº ØÖ Gß x vø\²à»õ ÒÎa A A A \õº GÛÀ GÚ { ÄP. (ii) r xiˆ yj ˆ zkˆ ØÖ r r GÛÀ. r 0, GÚU Põs UPÄ. r (iii) r xiˆ yj ˆ zkˆ ØÖ r r GÛÀ 1 logr r, GÚU Põs UPÄ. (i) If A is a vector point function, a scalar point function, then prove that A A A. (ii) Show that. r 0, r where r xiˆ yj ˆ zkˆ, r r. 1 (iii) Show that logr, r where r xiˆ yj ˆ zkˆ, r r. Page 1 ode No. : 08

14 w (B) (i) (ii) Gß x J vø\ À»õu ÒÎa \õº GÛÀ ß Q μi siß PºÀ øó² GÚU Põs UPÄ. r xiˆ yj ˆ zkˆ ØÖ r r GÛÀ f r f r f r Põs UPÄ. Page 1 r GÚU (i) If is a scalar point function, then prove that the curl of the gradient of variables. (ii) If r xiˆ yj ˆ zkˆ and r r, then 18. (A) Põmk öuõøp hà r show that f r f r f r f. A1, A dr ode No. : 08. öuõøp hà õøuø \õμõx UP ßÔ ø õu ØÖ õx õú Pmk õk J vø\ À»õ ÒÎ \õº, f GßÓ { uøúø {Áºzv ö\ Ásk GÚ {¹ UPÄ. Prove that the necessary and sufficient condition for the line integral f dr to be. A1, A independent of the path of integration is the existence of a scalar point function such that f.

15 (B) (i) F xyiˆ x y zkˆ GßÓ Âø\U PÍzvÀ J xpò x t 1, y t 1, z t GßÓ ÁøÍÁøμ À, 0,1 ÒÎ ¼ x,, 9 GßÓ ÒÎUS h ö μ BS Áø» Ð 1 GßÖU Põs UPÄ. (ii) Gß x 0, 0 GßÓ ÒÎ ¼ x (i) 1, Áøμ²ÒÍ ÁøÍÁøμ x t, y t ØÖ F xyiˆ y GÛÀ F. dr ß v ø U Psk iupä. Show that the work done in moving a particle in a field of force F, where F xyiˆ x y zkˆ along the curve x t 1, y t 1, z t from the point, 0,1 to,, 9 is -1. (ii) If F xyiˆ y and is the curve x t, evaluate y t from 0, 0 to 1, F. dr. Page 15 ode No. : 08

16 w 19. (A) A xiˆ y z kˆ GßÓ öáuhº x y, z 0, z Ø μ PøÍ GÀø»PÍõPU öpõsh E øí ÁiÁ õú μ hzvà Põì øháºásm uøózøu {Áºzv ö\ ² Gß øu \ õºupä. Verify the Gauss divergence theorem for A xiˆ y z kˆ taken over the cylindrical region bounded by the surfaces x y, z 0, z. (B) A yzi ˆ y xz kˆ GßÓ öáuhº x 0, y 0, z 0, z ØÖ x y 9 BQ ÁØøÓ GÀø»PÍõPU öpõskòí μ hzvß «x øháºásm uøózøu {Âºzv ö\ ² Gß øu \ õºupä. Verify the divergence theorem for A yzi ˆ y xz kˆ taken over the region bounded by x 0, y 0, z 0, z and x y 9. Page 16 ode No. : 08

17 0. (A) A xyiˆ yzj ˆ y kˆ GßÓ öáuhº x 0, y 0, z 0 GßÓ uí[pøí GÀø»PÍõPU öpõskòí x y z 1 GßÓ Ø μ Ýøh Áμ ß «x ì hõu uøózøu {Áºzv ö\ ² Gß øu \ õºupä. Verify Stokes theorem for A xyiˆ yzj ˆ y kˆ taken over the triangular surface S is the plane x y z 1 bounded by the planes x 0, y 0, z 0 over its boundary. (B) (i) Gß x x y 1 GßÓ Ámh GÛÀ, (ii) x ydx x dy GßÓ, öuõøp±k QŸß uøózøu {Áºzv ö\ ² Gß øu \ õºupä. A Miˆ N GÛÀ, uír ß uøózøu RÌPshÁõÖ GÊx i² Gß øuu Põs iupä A. dr. A. kdxdy ˆ R Page 17 ode No. : 08

18 w (i) Verify Green's theorem for x ydx x dy, where is the circle x y 1. (ii) Show that Green's theorem plane can be expressed in the form A. dr. A. kdxdy ˆ, where A Miˆ N. R Page 18 ode No. : 08

Introduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8

Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular