VEKTORANALYS. CURVILINEAR COORDINATES (kroklinjiga koordinatsytem) Kursvecka 4. Kapitel 10 Sidor

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1 VEKTORANALYS Kusvecka 4 CURVILINEAR COORDINATES (koklinjiga koodinatstem) Kapitel 10 Sido 99-11

2 TARGET PROBLEM An athlete is otating a hamme Calculate the foce on the ams. F ams F F ma dv a v dt d v dt eˆ + eˆ 0 0 (in clindical cood.) F ams F We need to intoduce cuvilinea coodinates to descibe clindical coodinates to calculate the deivative of ê 1

3 Clindical coodinates ae an eample of cuvilinea coodinates catesian cood. P: 0, 0, 0 CYLINDRICAL COORDINATES ê P eˆ 0 ê clindical cood. P:,, tan / cos sin 0 0 π +

4 CYLINDRICAL COORDINATE SYSTEM: infomal intoduction cos sin (,, ) eˆ ˆ ˆ + e + e ( cos, sin, ) coseˆ + sineˆ + eˆ (it is still in a catesian coodinate sstem) (,, ) (1,0,0) e This is the ate of vaiation of with keeping and constant it is a vecto paallel to the -ais (,, ) (0,1, 0) e This is the ate of vaiation of with keeping and constant it is a vecto paallel to the -ais How to calculate in a clindical coodinate sstem the e ais and the e ais? ( cos, sin, ) (cos,sin, 0) coseˆ sin ˆ + e e This is the ate of vaiation of ( cos, sin, ) ( sin, cos, 0) sineˆ cos ˆ + e e This is the ate of vaiation of eˆ with keeping and constant it is a vecto paallel to the -ais with keeping and constant it is a vecto paallel to the -ais eˆ P ê P 1 ê EXERCISE: Plot and in P e e EXERCISE: Epess in a clindical coodinate sstem (i.e. using ê and ê ) the vecto: v eˆ + eˆ 3

5 CURVILINEAR COORDINATE SYSTEMS Conside a catesian coodinate sstem,, The sufaces c c and c ae the coodinate sufaces. Thei intesection defines the -ais, -ais and -ais Conside anothe coodinate sstem defined b the vaiables u 1, u, u 3 u c u 3 u 1 c We assume that thee is a one-to-one elationship u 1 u 3 c u between i and u i, so that i can be epessed as a function of u i (and vice-vesa): - The sufaces defined b u i c ae called coodinate sufaces - The cuves defined b the intesection of the coodinate sufaces ae called coodinate cuves - The 3 cuves u 1, u, u 3 ae the coodinate aes u (, u, u) 1 3 u (, u, u) 1 3 u (, u, u) 1 3 DEFINITION A sstem of cuvilinea coodinates is othogonal if the coodinate cuves ae pependicula to each othe whee the intesect u u δ i j ik 4

6 Point P,, u 1, u, u 3 DEFINITION CURVILINEAR COORDINATES the basis is defined b the unit vectos: eˆ ˆ ˆ e e?? d ( d, d, d) eˆ eˆ eˆ An othogonal cuvilinea coodinate sstem has an othonomal basis { eˆ1, eˆ, eˆ3} in each point and 1 eˆi with scale facto h i h u u i i i u 3 ê 3 u ê ê 1 P (,,) u eˆ 1 magnitude 1 i hi ui hi ui Othonomal: eˆ ˆ i ej δij 1 1 othogonal eˆ ˆ i e j 0 fo i j h u h u i i j j d du + du + du h eˆdu + h eˆ du + h eˆ du u u u d h du eˆ i i i i Konecke delta 5

7 SURFACE ELEMENT AND VOLUME ELEMENT In a Catesian coodinate sstem: ( ) ˆ ˆ ˆ d d, d, d de + de + de ds ds ds dd dd dd dv ddd S S d S d d In a othogonal cuvilinea coodinate sstem: d h du eˆ + h du eˆ + h du eˆ u 3 ds ds ds h h du du h h du du h h du du dv h h h du du du u 1 S 3 S 1 S h 3 du 3 h du h 1 du 1 u 6

8 Othonomal basis CYLINDRICAL COORDINATES cos sin coseˆ + sineˆ + eˆ 1 eˆi with h i h u u i i i 0 ê P 0 eˆ 0 ê (cos,sin, 0) ( sin, cos, 0) (0, 0,1) h cos + sin 1 h ( sin) + ( cos) h 1 1 eˆ (cos,sin, 0) cos ˆ sin ˆ e + e h 1 eˆ ( sin, cos, 0) sin eˆ cos ˆ + e h 1 eˆ (0, 0,1) ˆ e h EXERCISE: wite using eˆ, ˆ, ˆ e e ds dv ds dd dd ddd 7

9 Fams F ma F dv a v dt d v dt eˆ + e 0 0ˆ TARGET PROBLEM (in clindical cood.) Fams ma mv m 0 (the length does not change) d d ( ˆ ˆ. ˆ ˆ. 0e + 0e + e 0 + e 0 ) dt dt. d eˆ ( cos,sin,0) ( sin, cos,0) ( sin,cos,0) eˆ dt. d eˆ ( 0, 0,1) 0 dt. d dt. d eˆ ( sin,cos,0) ( cos, sin,0) ( cos,sin,0) ( ) ( ) ( ) ( 0 eˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 e 0 e 0 e 0 e 0 e 0 e 0 e e) dt ( ) F ˆ ˆ ams m m 0 e e 0 (the hamme otates on a plane constant) eˆ 8

10 PRACTICAL EXAMPLE: THE BIOT-SAVART LAW The magnetic field in a point P of a stead line cuent is given b the Biot-Savat law: B() ( ') µ ' 0I dl 3 4 π ' L Whee dl ' is an infinitesimal length along the wie, I is the position vecto of the point P and dl ' ' is a vecto fom the oigin to dl ' Theefoe, ' is a vecto fom dl ' to P ' P ' 9

11 PRACTICAL EXAMPLE: THE BIOT-SAVART LAW The magnetic field in a point P of a stead line cuent is given b the Biot-Savat law: B() ( ') µ ' 0I dl 3 4 π ' L Calculate the magnetic field in P eˆ ˆ 0 + e 0 poduced b a staight wie along the -ais with cuent I and length b and cented at 0. SOLUTION: If is the distance fom the oigin to P, c in clindical coodinates: ( ) ˆ l( ') 0,0, ' e ' with ': b + b eˆ c dl ' d eˆ ' dl ' ' b -b ' eˆ c ' eˆ ' ' eˆ ' eˆ ' + ' c c ( ) ( ) dl ' ' d ' eˆ eˆ ' eˆ d ' eˆ c c b 0I b ˆ cd ' e 0I b d ' 0I ' 0I b ˆ ˆ ˆ b 3/ c b 3/ c π ( ' ) ( ' ) c c ' c π + c + π π + c b c + b µ µ µ µ B() e e e If the wie is infinitel long,we need to calculate the limit fo b B() µ I 0 eˆ π c 10

12 WHICH STATEMENT IS WRONG? ê 1- does not depend on the position (ellow) e - is constant evewhee (ed) ˆ 3- Clindical coodinates ae appopiate with a clindical smmet (geen) 4- Clindical coodinates ae a cuvilinea coodinate sstem (blue) 11

13 TARGET PROBLEM The electic field is consevative. The electostatic potential φ is defined b: E φ E φ E 0 fist Mawell s equation with no chage ( φ) φ 0 Laplace s equation Calculate the electostatic potential geneated outside a spheical chage. E Due to the spheical smmet, the solution will depend onl on the adius: φ φ() with + We need to: intoduce spheical coodinates calculate gadient and divegence in spheical coodinates solve the equation 1

14 spheical coodinates ae an eample of cuvilinea coodinates catesian cood. P: 0, 0, 0 SPHERICAL COORDINATES θ P 0 spheical cood. P:, θ, tanθ tan / 0 0 θ π 0 π 0 0 sinθcos sinθsin cosθ 13

15 Othonomal basis SPHERICAL COORDINATES ê ê θ P eˆ sinθcos sinθsin cosθ θ 0 sinθcoseˆ + sinθsineˆ + cosθeˆ 0 1 eˆi with h i h u u i i i 0 (sinθcos,sinθsin,cos θ) ( cosθcos, cosθsin, sin θ) θ ( sinθsin, sinθcos,0) h sin θcos + sin θsin + cos θ 1 hθ ( cosθcos) + ( cosθsin) + ( sinθ) h ( sinθsin) + ( sinθcos) sinθ 1 eˆ (sinθcos,sinθsin,cos θ) h 1 eˆ θ (cosθcos, cosθsin, sin θ) hθ θ 1 eˆ ( sin,cos,0) h EXERCISE: wite using eˆ, eˆ, eˆ ds sinθdθd θ dv sinθddθd 14

16 GRADIENT IN CURVILINEAR COORDINATES In catesian coodinates: φ φ φ φ φ φ gadφ,, eˆ ˆ ˆ + e + e And in a cuvilinea coodinate sstem? We must epess gadφ in tems of the cuvilinea basis eˆ, eˆ, eˆ : gadφ g eˆ + g eˆ + g eˆ since dφ gadφ d and d h du eˆ + h du eˆ + h du eˆ ( ˆ ˆ ˆ ) ( ˆ ˆ ˆ ) dφ g e + g e + g e h du e + h du e + h du e g h du + g h du + g h du φ φ φ dφ du + du + du u u u But also, witing φ as a function of u i : φ 1 φ 1 φ Theefoe: g1, g, g3 h u h u h u gadφ 1 h φ e u i i i ˆi 15

17 GRADIENT IN CURVILINEAR COORDINATES THE GRADIENT in clindical coodinates: 1 φ 1 φ 1 φ φ 1 φ φ gadφ eˆ eˆ eˆ ˆ ˆ ˆ + + e + e + e h h h in spheical coodinates: 1 φ 1 φ 1 φ φ 1 φ 1 φ gadφ eˆ ˆ ˆ ˆ ˆ ˆ + e + e e + e + e h h θ h θ sinθ θ θ θ EXERCISE: calculate in spheical coodinates 1 DIVERGENCE IN CURVILINEAR COORD. 1 diva A h h + A h h + A h h hhh u u u ( ) ( ) ( ) Poof: see theoem 10.4, page 109 CURL IN CURVILINEAR COORD. heˆ heˆ heˆ ota hhh u u u ha ha ha EXERCISE: calculate in spheical coodinates A Poof: see theoem 10.5, page 11 16

18 Due to spheical smmet φ φ() φ 0 Which can be witten as: ( φ) φ 0 TARGET PROBLEM with Due to spheical smmet, the solution is eas in spheical coodinates + E φ 1 φ 1 φ φ 1 φ 1 φ gadφ,, eˆ ˆ ˆ + eθ + e θ sinθ θ sinθ 1 1 diva ( A1h h3) ( Ah 3h1) ( A3h 1h) + + ( A sinθ) + ( Aθ sinθ) + ( A) hhh 1 3 u1 u u3 sinθ θ div( gadφ) θ + θ + + sinθ θ θ sinθ 1 φ 1 φ 1 φ φ φ sin sin 0 0 φ φ() No θ and no dependence φ φ c + 0 φ() + d c E gadφ e ˆ 17

19 WHICH STATEMENT IS WRONG? 1- The scale facto is necessa to calculate the gadient (ellow) - The scale facto is necessa to calculate the divegence (ed) 3- Spheical coodinates ae a cuvilinea coodinate sstem (blue) 4- In spheical coodinates the position vecto is (geen) (, θ, ) 18