ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]

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Lillian Bates
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1 ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,, ). [Noe ha his iangle is eniely in he plane z =.] (a) how ha he uni nomal ˆn o A ha poins away fom he oigin is nˆ k. ˆ (b) A veco fiel F is efine in F y ˆi x ˆj z x k ˆ. how ha cul ˆ x y by F j k ˆ. (c) Fin he ciculaion I F of F aoun. [Hin: you may use okes heoem.] (a) Any plane paallel o he xy plane has a uni nomal k ˆ. z = >. Theefoe he uni nomal poining away fom he oigin is nˆ k. ˆ OR PQ T i ˆ an QR T ˆj n ˆi ˆj kˆ (b) ˆ i y x ˆ F j x ˆ x y ˆ y j k x y ˆ k z x z

2 ENGI 44 Poblem e 9 oluions Page of (c) Using okes heoem, ˆ I F F n x y A A Bu aea A is a iangle, no a ecangle. x y x y I x y y x xy y x x x x x x x I OR Diec evaluaion Pah PQ: an Q F F I F P Pah QR: an F F R Q I F Pah RP:

3 ENGI 44 Poblem e 9 oluions Page of (c) (coninue) an F F P I F R I F I I I

4 ENGI 44 Poblem e 9 oluions Page 4 of. A veco fiel F is efine in he cylinical pola cooinae sysem in by F ˆ 5 Fin he oal flux F ue o F ou hough he igh cicula cyline of aius 5, heigh 6, aligne along he z-axis wih he op an boom ens a z an z, as shown. is a simple close suface. The veco fiel F has no singulaiies on o wihin (o inee anywhee in ). Theefoe we may employ Gauss ivegence heoem. [Don foge he scale facos in F!] iv ˆ F V F iv F V V 5 5 V V whee V is he volume enclose by he igh cicula cyline of aius 5 an heigh 6, V h V Theefoe OR evaluaing he suface inegal iecly, On he op suface, N kˆ F N On he boom suface, N kˆ F N The ouwa uni nomal o he cuve suface is eveywhee N ˆ 5 Eveywhee on he cuve suface 5 F ˆ ˆ 5 Use he cylinical pola cooinaes, z on he cuve suface. The elemen of suface aea hee is A z 5 z sie F A z z z sie sie op ˆ ˆ bo 5 6 an op bo sie 6 6.

ENGI 44 Poblem e 9 oluions Page 5 of. An exene souce of elecic chage has a chage ensiy e whee = he isance of he poin (x, y, z) fom he oigin. (a) Fin he oal chage Q ue o his exene objec.

5 ENGI 44 Poblem e 9 oluions Page 5 of. An exene souce of elecic chage has a chage ensiy e whee = he isance of he poin (x, y, z) fom he oigin. (a) Fin he oal chage Q ue o his exene objec. The chage is non-zeo on an wihin a sphee of aius, cene he oigin. Theefoe wok in spheical pola cooinaes. Q V e sin V sin e [inegaion by pas ] 8e 6 cos 6 6 e Q e (b) Fin he oal flux ue o he exene chage hough he simple close suface efine by x y 4 z. is a sphee, cene (, 4, ), aius. The cene of is 5 unis away fom he oigin O All pas of ae heefoe a leas 4 unis away fom O The chage exens no moe han unis away fom O. Theefoe he suface encloses no chage a all. Fom Poisson s equaion iv E eveywhee insie Using Gauss ivegence heoem, Flux E E V Theefoe he oal flux hough is V

6 ENGI 44 Poblem e 9 oluions Page 6 of 4. alculae he ciculaion of F x y x y x y z e couneclockwise aoun he uni cicle in he xy-plane. [Hin: Use okes heoem, leing he suface be any smooh suface ha has as is bounay.] xyz T hoose o be he ineio of he uni cicle in he x-y plane. The omain of F is all, so okes heoem is vali. F F ˆ i x y x 4 xyz x yz x y z e ˆ xyz cul F j x y x y z x y z e y xy ˆ xyz k x y z e z Bu on, z = cul F xy ˆ k eveywhee on uface ne meho The cicula isk suggess he paamees, such ha cos sin T ˆi cos sin N ˆj sin cos kˆ kˆ hoose he ouwa nomal o be poining up ou of he x-y plane in he iecion of inceasing z. Then N kˆ k ˆ an F xy kˆ kˆ cos sin sin F sin 4 sin sin 4 4 cos F

7 ENGI 44 Poblem e 9 oluions Page 7 of 4 (coninue) OR Pojecion meho In his case, he pojecion is ivial, as he enie suface is aleay on he x-y plane. Fomally, z z z n x y T z z an n ˆ ˆ A x y k k Bu is he ineio of a cicle, no a ecangle, so use plane pola cooinaes. A k ˆ an F xy kˆ kˆ cos sin sin F sin 4 sin sin 4 4 cos F OR [ignoing he hin]

8 ENGI 44 Poblem e 9 oluions Page 8 of 4 (coninue) The line inegal can be evaluae iecly. On an obvious paameeizaion is x = c, y = s, z =, whee c = cos an s = sin F c s c s T c s s c Bu sin cos T T F cs s c s cos cs c s F F Now cos sin c s 4 c c so ha cs c s c c s c c c 4 cos sin an moe obviously 4 cos sin cos F F

ENGI 44 Poblem e 9 oluions Page 9 of 5. omplee he evaluaion (in Example.4 of he lecue noes) of he line inegal F F aoun he uni squae in he x-z plane fo xy xyz xz e T, (wihou using okes heoem).

9 ENGI 44 Poblem e 9 oluions Page 9 of 5. omplee he evaluaion (in Example.4 of he lecue noes) of he line inegal F F aoun he uni squae in he x-z plane fo xy xyz xz e T, (wihou using okes heoem). an Fom example.4, he posiive oienaion aoun he squae is OGHJ (he y axis is iece ino he page). In he x-z plane y = T F xz ompue he line inegal aoun he fou sies of he squae, saing wih OG: OG : ˆ k F ˆ ˆ F k k OG F The above was pesene in a lecue. Below ae he calculaions fo he ohe hee sies of he squae: GH : ˆ i an F F F GH

10 ENGI 44 Poblem e 9 oluions Page of 5 (coninue) HJ : ˆ k an F F HJ F JO : ˆ i an F F F JO Aing he fou segmens ogehe F F F F F Theefoe OG GH HJ JO F

11 ENGI 44 Poblem e 9 oluions Page of 6. A veco fiel F is efine in he spheical pola cooinae sysem in F e ˆ (a) Fin he ivegence F in spheical pola cooinaes. (b) Fin he oal flux F ue o F ou hough he sphee of aius, cene a he oigin. by (a) Fo spheical pola cooinaes geneally, sin f sin f f sin F Howeve, F is puely aial F sin f f sin e e F F e (b) Thee ae no singulaiies of F o is eivaives anywhee in Gauss ivegence heoem may be use. F F V e sin V [Noe ha F canno be exace fom insie he iple inegal because i is no consan insie he volume V.] sin e Afe an inegaion by pas, cos e 8e + 6 e 6 6 e + 6 e e e OR e

12 ENGI 44 Poblem e 9 oluions Page of 6 (b) (coninue) Evaluaing he ouble inegal iecly, by he suface ne meho, a paameic ne fo he suface of he sphee is sincos sinsin cos ˆi cos cos sin sin N ˆj cos sin sin cos kˆ sin 4sin cos sin cos N 4sin sin sin sin sin sin 4sin cos cos sin cos (o one may quoe he nomal veco fo he sphee of aius a cene he oigin: N asin ) Eveywhee on. The ouwa nomal is N sin 4sin ˆ. 8 F N e ˆ 4sin ˆ sin e 8 8 F sin sin e e 8 8 cos e e e OR Using he pojecion meho, he uppe an lowe hemisphees nee o be consiee sepaaely. ymmey he flux hough each hemisphee will be equal. z f x, y 4 x y The equaion of he uppe hemisphee is z x x z y an, by symmey, x 4x y z y z

13 ENGI 44 Poblem e 9 oluions Page of 6 (b) (coninue) z ˆ N z ˆ ˆ x ˆ y ˆ ˆ x ˆ y ˆ z ˆ ˆ x i y j k z i z j k z i j k z z The shaow of he hemisphee on he xy-plane is he cicle of aius, cene he oigin. e F ˆ bu x y z 4 F N e ˆ ˆ 4 4 x y z e z e z D [The faco of accouns fo boh hemisphees.] wich o plane pola cooinaes, hen an x y z 4 x y e 4 e e e e Back o he inex of soluions

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