φ (x,y,z) in the direction of a is given by

Transcription

1 UNIT-II VECTOR CALCULUS Dectoal devatve The devatve o a pot ucto (scala o vecto) a patcula decto s called ts dectoal devatve alo the decto. The dectoal devatve o a scala pot ucto a ve decto s the ate o chae o the decto. It s ve b the compoet o ad that decto. The dectoal devatve o a scala pot ucto (,,) the decto o a s ve b.a. a Dectoal devatve o s mamum the decto o Hece the mamum dectoal devatve s oad. Ut omal vecto to the suace I (,, ) be a scala ucto, the (,, ) c epesets A suace ad the ut omal vecto to the suace s ve b Equato o the taet plae ad omal to the suace Suppose a s the posto vecto o the pot,, ) O the suace (,, ) c. I ( s the posto vecto o a pot (,,) o the taet plae to the suace at a, the the equato o the taet plae to the suace at a ve pot a o t s ve b a. ad I s the posto vecto o a pot o the omal to the suace at the pot a o t. The vecto equato o the omal at a ve pot a o the suace s a ad The Catesa om o the omal at,, ) o the suace ( (,,) c s o Dveece o a vecto I (,, ) s a cotuousl deetable vecto pot ucto a ve eo o space, the the dveeces o s deed b. dv

2 I,the ).( dv.e., dv Soleodal Vecto A vecto s sad to be soleodal dv (e). Cul o vecto ucto I ),, ( s a deetable vecto pot ucto deed at each pot (,, ), the the cul o s deed b cul I,the ) ( cul cul Cul s also sad to be otato Iotatoal Vecto A vecto s called otatoal Cul (e) Scala Potetal I s a otatoal vecto, the thee ests a scala ucto Such that. Such a scala ucto s called scala potetal o Popetes o Gadet. I ad ae two scala pot ucto that ( ) ± ± (o) ( ) ad ad ad ± ± Soluto: ( ) ( ) ± ±

3 ( ) ( ) ( ) ± ± ± ± ± ± ± ±. I ad ae two scala pot uctos the ( ) (o) ad ad ad ) ( Soluto: ( ) ( ) ( ) ( ) ( ). I ad ae two scala pot ucto the whee Soluto: [ ]. I such that,pove that Soluto:

4 5. d a ut omal to the suace at (,-, ) Soluto: Gve that ) ( ( ) ( ) ( ) At (,-, ) ( ) ) ( () Ut omal to the ve suace at (,-,) 6 6. d the dectoal devatve o at (,,) the decto o Soluto: Gve ) ( ( ) ( ) ( ) 8 At (,, ) Gve: a 6 a

5 a a D D [ ] [ ] d the ale betwee the suace 5 ad 5 at (,,) Soluto: Let ad,,,, ) ( At (o,,) Cos θ 6 cos θ cos θ cos 8. d the ale betwee the suaces lo ad at the pot (,,) Soluto: let lo ad,, lo,, ) lo (

6 Cos 6 5. θ 6 5 cos θ 9. d ( ) Soluto: ( ) ( ). ( ) ( ) ( ) ( ) Sce u dv u u. ( ) ( )... ( ) ( ). ( )( ). ( )( ) ( ) [ ] ( )

7 . I ad.pove that s soleodal ad s otatoal o all vectos o. Soluto: dv ( ) ( ) ( ) () Now Deetat patall w..to, Smlal, Now ( ) ( ).. ( ) ( ) om () we have ( ) dv ( ) The vecto s soleodal dv ( ) s soleodal ol - Now cul ( ) ( )

8 ( ) Cul ( ) Cul ( ) o all values o Hece s otatoal o all values o.. Pove that ( ) ( ) s cos s otatoal ad d ts scala potetal Soluto: s cos cul [ ] [ ] [ ] cos cos s otatoal. To d such that ad ( ) ( ) s cos Iteat the equato patall w..to,, espectvel ), s ( ), ( s ), (, s C s scala potetal. Pove that ).( ).( B cul A A cul B B A dv Poo : ).( B A B A dv B A B A B A B A A B

9 B A. A. B cul B. A cula. B.Pove that Soluto: culcul cul cul B us a b c a. c b a. b c.. ( ). VECTOR INTEGRATION Le, suace ad Volume Iteals Poblems based o le Iteal Eample : I ( 6) (,,) alo the cuve Evaluate. d om (,,) to t, t, t Soluto: The ed pots ae (,, ) ad (,, ) These pots coespod to t ad t d dt, d t, d t. d ( 6) d d d C C 5 7 ( t 6t ) dt t ( tdt) t ( t ) dt 6 9 ( t 8t 6t ) 9 dt t t 6t 7 ( ) [( 6) ] 5 C Eample : Show that s a cosevatve vecto eld.

10 Soluto: I s cosevatve the s a cosevatve vecto eld. Now Suace Iteals Deto: Cosde a suace S. Let deote the ut outwad omal to the suace S. Let R be the poecto o the suace o the XY plae. Let be a vecto valued deed some eo cota the suace S. The the suace teal o s deed to be Eample ; Evaluate S S.. ds d. d R.. ds whee ad S s the suace o the clde cluded the st octat betwee the plaes ad. Soluto: Gve The ut omal to the suace..

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14 () () () INTEGRAL THEOREMS Gauss s dveece theoem Stoe s theoem Gee s theoem the plae Gee s Theoem Statemet: I M(,) ad N(,) ae cotuous uctos wth cotuous patal devatves a eo R o the plae bouded b a smple closed cuve C, the N M Md d dd, whee C s the cuve descbed the c R postve decto.

15 Ve Gee s theoem a plae o the teal ( ) d d tae aoud the ccle Soluto: Gee s theoem ves N M Md Nd dd c R Cosde ( ) d d c M N M N, N M dd R dd dd ( ) R R [Aea o the ccle] π. π. π () Now Md Nd We ow that the paametc equato o the ccle cosθ sθ d sθdθ, d cosθdθ Md Nd ( ) d d cosθ sθ sθdθ cosθ cosθ d ( )( ) ( ) θ cosθ sθ 8s θ cos θdθ Whee θ vaous om to π π ( cosθ sθ s θ ) Md Nd dθ C π cos θ s θ d θ π ( s θ 6 cos θ ) d θ cos θ s θ 6θ π π.() om () ad () π c

16 N M Md Nd dd c R Hece Gee s Theoem s veed. Eample Us Gee s theoems d the aea o a ccle o adus. Soluto: B Gee s theoem we ow that Aea eclosed b C d d The paametc equato o a ccle o adus s Whee θ π π C Aea o the ccle cosθ ( cosθ ) sθ ( sθ ) dθ Eample : π ( cos θ s θ ) dθ π dθ θ π π [ ] Evaluate [( ) d cosd] c cosθ, sθ s whee c s the tale wth π vetces (,),(,) ad ( π,) Soluto: Equato o OB s π π

17 N M B Gee s theoem Md Nd dd c R M Hee M s, N N cos, s [( s ) d cosd] ( s )dd C I the eo R, vaes om C ( s ) d cosd ( s ) R π π [ cos ] π π to ad vaes om to dd π π d π π π cos d π π π s π π π π π π Eample Ve Gee s theoem the plae o 8 d 6d whee C s the bouda o the eo deed ( ) ( ) C b X,, Soluto: We have to pove that

18 c N M Md Nd dd R M 8, N 6 M N 6, 6 B Gee s theoem the plae N M Md Nd dd c R ( ) dd ( ) 5 d ( ) 5 Cosde Md Nd c OA Alo OA,, vaes om to OA AB Md Nd BO 5 d [ ] Alo AB, - d d ad vaes om to. AB STOKE S THEOREM Md Nd [ 8( ) ( ) 6( ) ]d ( ) ( ) 8 8 8

19 I S s a ope suace bouded b a smple closed cuve C ad a vecto ucto s cotuous ad has cotuous patal devatves S ad o C, the cul. ds. d whee s the ut vecto omal to the c suace (e) The suace teal o the omal compoet o to the teal o the taetal compoet o tae aoud C. Eample Ve Stoe s theoem o ( ) cul s equal whee S s the uppe hal o the sphee ad C s the ccula bouda o plae. Soluto: B Stoe s theoem Hee ( ) c cul. d cul. ds [ ] ( ) ( ) s sce C s the ccula bouda o plae S aea o the ccle cul. ds S dd π ( ) π.() ON, c O C, cosθ, sθ d s θdθ, d cosθdθ θ vaes om to π. d cul. ds s

20 c. d π om () ad () Eample ( cosθ sθ )( sθ ) dθ c π π ( cosθ sθ ) d θ π s θdθ π ( s θ ) d θ π cos θ dθ π cos θ s θ θ π π (). d cul. ds Hece stoe s theoem s veed s Ve stoe s theoem o ( ) ( ) whee s s the suace o the cube,,,, ad above the plae. Soluto: B Stoe s theoem c. d cul. ds Gve ( ) ( ) s cul [ ] [ ] [ ] [ ]

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22 Hece Stoe s theoem s veed. Eample : Ve Stoe s theoem o whee S s the uppe hal suace o the sphee ad C s ts bouda. Soluto: B stoe s theoem

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25 c. d cul. ds s Gauss Dveece theoem Statemet: The suace teal o the omal compoet o a vecto ucto ove a closed suace S eclos volume V s equal to the volume teal o the dveece o tae thouhout the volume V, S. ds. dv V Evaluate dd dd dd ove the suace bouded b, h, a Soluto:

26 π π cos θd θ 6 π S a. ds

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