Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats. Ths ar two important xampls of what ar calld curvilinar coordinats. In this lctur w st up a formalism to dal with ths rathr gnral coordinat systms. Actually w hav ffctivly covrd much of th matrical in lcturs 8 and 9 whr w studid surfac and volum intgrals. Thr w considrd paramtrisation of surfacs and volums. Hr w do th sam thing but think about how this raliss non-cartsian Coordinat systms. Suppos w chang from th Cartsian coordinats (x, x 2, x 3 to th curvilinar coordinats, which w dnot u i, ach of which ar functions of th x i :- u = u (x, x 2, x 3 u 2 = u 2 (x, x 2, x 3 u 3 = u 3 (x, x 2, x 3 Th u i should b singl-valud, xcpt possibly at crtain points, so th rvrs transformation, x i = x i (u, u 2, u 3 can b mad. A point may b rfrrd to by its Cartsian coordinats x i, or by its curvilinar coordinats u i. For xampl, in 2-D, w might hav:- Now considr coordinat surfacs dfind by kping on coordinat constant. Th Cartsian coordinat surfacs x i = constant ar plans, with constant unit normal vctors i (or, 2 and 3, intrscting at right angls. Th surfacs u i = constant do not, in gnral, hav constant unit normal vctors, nor in gnral do thy intrsct at right angls. 89
Exampl: Sphrical polar co-ordinats r = { } x 2 y 2 z 2 ; θ = cos z x2 y 2 z 2 ; φ = tan ( y x. Th surfacs of constant r, θ, and φ ar:- r = constant = sphrs cntrd at th origin unit normal r θ = constant = cons of smi-angl θ and axis along th z-axis unit normal θ φ = constant = plans passing through th z-axis unit normal φ Ths surfacs ar not all plans, but thy do intrsct at right angls. If th coordinat surfacs intrsct at right angls (i.. th unit normals intrsct at right angls, as in th xampl of sphrical polars, th curvilinar coordinats ar said to b orthogonal. 23.. Orthogonal Curvilinar Coordinats Unit Vctors and Scal Factors Suppos th point P has position r = r(u, u 2, u 3. If w chang u by a small amount, du, thn r movs to position (r dr, whr dr = u du h du whr w hav dfind th unit vctor and th scal factor h by h = u and = h u. Th vctor is a unit vctor in th dirction of incrasing u. Th scal factor h givs th magnitud of dr whn w mak th chang u u du. Thus for an infinitssimal chang of u dr = h du 90
Similarly, w can dfin h i and i for i = 2 and 3. Th unit vctors i ar not constant vctors. In gnral thy ar non- Cartsian basis vctors, thy dpnd on th position vctor r, i.. thir dirctions chang as th u i ar varid. If i j = δ ij, thn th u i ar orthogonal curvilinar coordinats. For Cartsian coordinats, th scal factors ar unity and th unit vctors i rduc to th Cartsian basis vctors w hav usd throughout th cours: r = x y 2 z 3 so that h = x =, tc. Exampl: sphrical polars: u = r, u 2 = θ and u 3 = φ in that ordr: r = r sin θ cos φ i r sin θ sin φ j r cos θ k, whr to avoid confusion in this sction w us i, j, k for th Cartsian basis vctors. (By this stag thr should b no confusion with th suffics i, j, k. Thrfor Thus = sin θ cos φ i sin θ sin φ j cos θ k = hr = θ = r cos θ cos φ i r cos θ sin φ j r sin θ k = h θ = r φ = r sin θ sin φ i r sin θ cos φ j = h φ = r sin θ r = sin θ cos φ i sin θ sin φ j cos θ k = r / r θ = cos θ cos φ i cos θ sin φ j sin θ k φ = sin φ i cos φ j Ths unit vctors ar normal to th lvl surfacs dscribd abov (sphrs, cons and plans and ar clarly orthogonal: r θ = r φ = θ φ = 0 and form a RH orthonormal basis: r θ = φ, θ φ = r, φ r = θ. Exampl: Cylindrical polars: u = ρ, u 2 = φ and u 3 = z in that ordr: Thus and Thrfor r = ρ cos φ i ρ sin φ j z k. ρ = cos φ i sin φ j ; φ = ρ sin φ i ρ cos φ j ; z = k hρ = ; h φ = ρ ; hz = ρ = cos φ i sin φ j ; φ = sin φ i cos φ j ; z = k 9
Ths unit vctors ar normal to th lvl surfacs dscribd by cylindrs about th z-axis (ρ = constant, plans through th z-axis (φ = constant, plans prpndicular to th z axis (z = constant and ar clarly orthogonal. Rmark: An xampl of a curvilinar coordinat systm which is not orthogonal is providd by th systm of lliptical cylindrical coordinats (s tutuorial 9.4. r = a ρ cos θ i b ρ sin θ j z k (a b In th following w shall only considr orthogonal systms Arc Lngth Th arc lngth ds is th lngth of th infinitsimal vctor dr :- In Cartsian coordinats (ds 2 = dr dr. (ds 2 = (dx 2 (dy 2 (dz 2. In curvilinar coordinats, if w chang all thr coordinats u i by infinitsimal amounts du i, thn w hav dr = u du du 2 du 3 = h du h 2 du 2 2 h 3 du 3 3 For th cas of orthogonal curvilinars, bcaus th basis vctors ar orthonormal w hav For sphrical polars, w showd that (ds 2 = h 2 du 2 h 2 2 du 2 2 h 2 3 du 2 3 hr =, h θ = r, and h φ = r sin θ thrfor (ds 2 = (dr 2 r 2 (dθ 2 r 2 sin 2 θ (dφ 2 92
Lctur 23 continud: 23. 2. Elmnts of Ara and Volum Basically w just rpat using scal factors what w did in lcturs 8 and 9. Vctor Ara If u u du, thn r r dr, whr dr = h du, and if u 2 u 2 du 2, thn r r dr 2, whr dr 2 = h 2 2 du 2. (Curvatur gratly xaggratd in figur! On th surfac of constant u th vctor ara boundd by dr 2 and dr 3 is givn by ds = (dr 2 (dr 3 = (h 2 du 2 2 (h 3 du 3 3 = h 2 h 3 du 2 du 3, sinc 2 3 = for orthogonal systms. Thus ds is a vctor pointing in th dirction of th normal to th surfacs u =constant, its magnitud bing th ara of th small paralllogram with dgs dr 2 and dr 3. Similarly, on can dfin ds 2 and ds 3. For th cas of sphrical polars, if w vary θ and φ, kping r fixd, thn ds r = ( h θ dθ θ ( hφ dφ φ = hθ h φ dθ dφ r = r 2 sin θ dθ dφ r. Similarly for ds θ and ds φ. Volum Th volum containd in th paralllpipd with dgs dr, dr 2 and dr 3, is dv = dr dr 2 dr 3 = (h du (h 2 du 2 2 (h 3 du 3 3 = h h 2 h 3 du du 2 du 3 bcaus 2 3 =. For sphrical polars, w hav dv = hr h θ h φ dr dθ dφ = r 2 sin θ dr dθ dφ 93
Componnts of a Vctor Fild in Curvilinar Coordinats A vctor fild A(r can b xprssd in trms of curvilinar componnts A i, dfind as: A(r = i A i (u, u 2, u 3 i whr i is th ith basis vctor for th curvilinar coordinat systm. For orthogonal curvilinar coordinats, th componnt A i is obtaind by taking th scalar product of A with th ith (curvilinar basis vctor i A i = i A(r NB A i must b xprssd in trms of u i (not x, y, z whn working in th u i basis. Exampl If A = i in Cartsian coordinats, thn in sphrical polars,ar = A r = sin θ cos φ, tc. A(r, θ, φ = sin θ cos φ r cos θ cos φ θ sin φ φ. Exampl If A = r thn in cylindrical polars Aρ = A ρ = ρ cos 2 φ ρ sin 2 φ = ρ tc, and A(ρ, φ, z = r = ρ ρ z z 23. 3. Grad, Div, Curl, and th Laplacian in Orthogonal Curvilinars W dfind th vctor oprators grad, div, curl firstly in Cartsian coordinats, thn most gnrally through intgral dfinitions without rgard to a coordinat systm. Hr w complt th pictur by providing th dfinitions in any orthogonal curvilinar coordinat systm. Gradint In sction (2 w dfind th gradint in trms of th chang in a scalar fild f whn w lt r r dr δf = f(r dr ( Now considr what happns whn w writ f in trms of orthogonal curvilinar coordinats f = f(u, u 2, u 3. As bfor, w dnot th curvilinar basis vctors by, 2, and 3. Lt u u du, u 2 u 2 du 2, and u 3 u 3 du 3. By Taylor s thorm, w hav W hav alrady shown that δf = du du 2 du 3 u dr = h du h 2 du 2 2 h 3 du 3 3. 94
Using th orthogonality of th basis vctors, i j = δ ij, w can writ δf = = = ( u 2 ( du 2 du 2 3 du 3 3 ( h u h 2 2 h 3 (h du h 2 2 du 2 h 3 3 du 3 3 ( h u h 2 2 h 3 3 Comparing this rsult with quation ( abov, w obtain th following xprssion for f in orthogonal curvilinars dr f = h u h 2 = 3 i= h i u i i 2 h 3 3 For sphrical polars, w obtain f(r, θ, φ = r θ For cylindrical polars, w obtain r f(ρ, φ, z = ρ ρ φ θ φ ρ r sin θ φ z z. φ. Divrgnc In orthogonal curvilinar coordinats diva = h h 2 h 3 { u (A h 2 h 3 (A 2 h 3 h } (A 3 h h 2 This xprssion can b obtaind by using th intgral dfinition of diva, or altrnativly using vctor oprator idntitis (s BK 4.3, RHB 8.0. For Cartsian coordinats, w hav h i =, and w rgain th usual xprssion for A in Cartsians. For sphrical polars w hav diva(r, θ, φ = r 2 sin θ { ( r 2 sin θar θ = ( r 2 Ar r 2 r sin θ 95 ( ( } r sin θaθ raφ φ { ( ( } sin θaθ Aφ. θ φ
whr Ar, A θ, and A φ ar th componnts of th vctor fild A in th basis (r, θ, φ. For cylindrical polars w hav diva(ρ, φ, z = ρ { ( ρa ρ ρ φ A φ } z (ρa z = ( ρa ρ ρ ρ ρ φ A φ z A z. whr Aρ, A φ, and Az ar th componnts of th vctor fild A in th basis (ρ, φ, z. Curl In orthogonal curvilinar co-ordinats, curl is most convnintly writtn as A = h h 2 h 3 h h 2 2 h 3 3 u h A h 2 A 2 h 3 A 3.g. th first componnt is givn by A = h 2 h 3 { (A 3 h 3 } (A 2 h 2 and th componnts of A in th 2 and 3 dirctions may b obtaind by cyclic prmutations of th suffics. Th abov formula can b dmonstratd by using th lin intgral dfinition of curl, as usd to proof Stoks thorm, (s tutorial or by vctor oprator idntitis (BK. 4.3 or RHB 8.0. For sphrical polars w hav A = r r θ r sin θ φ r 2 sin θ θ φ A r ra θ r sin θ A φ Laplacian of a Scalar Fild Th Laplacian oprator acting on a scalar fild is dfind by 2 f = ( f, giving: 2 f = { ( h2 h 3 h h 2 h 3 u h u ( h3 h h 2 ( h h 2 h 3 } In sphrical polars, w hav 2 f(r, θ, φ = { ( r 2 sin θ ( sin θ ( r 2 sin θ θ θ φ sin θ = ( r 2 { r 2 r 2 sin 2 sin θ ( sin θ } 2 f θ θ θ φ 2 96 } φ.