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Coordinate Systems Operation Cartesian coordinates (x.y.z) Cylindrical coordinates (p,<p,z) Spherical coordinates (r,0,<p) Definition of coordinates p = s/x2 + y2 <f> = arctan(j//x) z = z x = pcos<f> y = psin<j) z = z X = r sin 9 COS < y = r sin 9 sin <; z = r cos 9 r - \Jx2 + y2 + z-9 = arccos(z/r) <j> = arctan(y/x-) r - \J(P- + z2 9 = arctan (p/ z) p = rsin(0) <t> = 4> z = rcos(0) Definition of unit vectors <j) — — 2x + £y v p pJ X = COS <t>p — sin 4><j> ÿ = sin <pp + cos 4><j> z = z X = sin 9cOS(pf + COS0COS00 — sin <p</> ÿ = sin 9 sm 4>f + cos 9 sin <pd + cos <p<t> z = cos 9r — sin 90 - _ xx+yÿ + ïî Q _ I-.X+yZy~p-i rp 0 _ -y*+xy f = -Ô+ ‘-i r’ r 0 = ÎP-JÎ (¡> = <t> p = sin Ov + cos 90 4> = 4> z = cos 9r — sin 90 A vector field A .4xx + .4yy + .4;z App + A^4> + A-z Arf + Ae0 + .-1 Gradient V f df. df. 9/. feX+%y + öTZ df dp pd<(> dzZ df*±ldfâ + or r Ou rsin Divergence V-A a.4, dAy dA. dx dz 13 (/>.4,) 13.4* 9.4, dp p d<t> + dz 1 3 (r2.4r) 1 3r r sin 0 89 (Ag sill#) + r sin 0 d<t> curl V x A 9.4; dAy\ 1 Ä , * dz J dAx dAz\ dz dx J ! y 'dAy dAx \ 7 dx dy y r fldAz dAA \P d<(> dz ) (%-%)* V 1 fd(pA^) _ dAA p V dp d<t> ) i( _L £ ‘w), r \sm 0 d(j> dr ) 1 ( 0 . BAA , r + Laplace operator A / = V2/ d2f 32/ &f dx2 %2 022 ii.f ÊL\ + 1^L + ËL pdp v dp) p1 dtp2 dz2 U(Æ\+ > r2 dr \ dr J r2sin 0 30 3 / . „3/ 99181"9» V > y r2sill2 ö 902 Vector Laplacian AA = V2A A.4*x + A.4yy + A.4;z , Ap 2 c).4„ _ K ' p p2 p2 d<t> P A* 2 dAp P2 + p2 d4> (A.4;) z A i 2Ar 2 9(^osin0) “ 13- “ —ä«— A J .4 a , 2 2 cos 0 dAô\ A 0 r- sin2 0 ~r^ ()0 r- sin2 0 ()p J ( a 1 -*ô . 2 I 2 cos 0 î y ^ r2 sin2 Ö r2s»nô ¿Î6 r2 sin2 ô f)o y ^ Differential displacement rfl = Æcx + rfi/ÿ + dzz dl = dpp + pdcpcj) + dzz c/l — drr + rdOO + r sin dd<p<j> Differential normal area dS = dy dz x+ dx dz ÿ+ dx dy z dS p d<}> dz p+ dp dz 4>+ pdpdéz dS = r2 sin 9 d9 d<t> r+ r sin 9 drd<pO-\-rdrddcj) Differential volume dV = dxdy dz dV = pdpd(f>dz dV = r sin 9 dr dd d<p — = - sin ¡fix + cos y - ip 9<fi .... ■S— = — COSwX — Slll s y = — S ap %- df_ difi df dd dê dip : - Si11 0 Sill px + sin 9 COS W = Sin Oê » = _ sin g CQS ^ ^ shl ^ - _ CQS d£ = _ . ¿/0 cos 0 cos V5X + cos e sill pÿ - sin dz = 0 ^ = - cos v?x - sin pÿ = -sin0r - cos 00 — = — cos0sin<^x +COSÖCOS(^y = cosô<^ Vector CatcuCus A A = A A* = |A| A + B = B + A A B = B-A A x B = -B x A (A + B) • C = A • C + B • C (A + B)xC = AxC + BxC A • (B x C) = B • (C x A) = C ■ (A x B) Ax (B x C) = (A • C) B - (A • B) C (A x B) • (C x D) = (A • C) (B • D) - (B • C) (A ■ D) (A x B) x (C x D) = (A • B x D) C — (A • B x C) D V x (A x B) = AV • B - BV • A + (B • V) A - (A • V) B V (A • B) = (A • V) B + (B • V) A + A x (V x B) + B x (V x A) v(V+<f>) = v^+V0 V(*/>0) = <j> + ip V0 V • (A + B) = V ■ A + V • B V x (A + B) = V x A + V x B V • {ipA) = vV • A + A • Vip V x (ipA) = vV x A — A x Vip V • (A x B) = B • V x A — A ■ V x B V • (V x A) = 0 f Ho Idl x r J '■ 47r |7'|3 E — —V<fi V2^ = S<) V x (V^1) = 0 V • (V^) = V2# V (V • A) - V x V x A = V2A (v’-'“£)e (v-/..£)b V • D = p{ V • B = 0 V x E= - m cH VxH = J/+f V • E = — ^0 <Jlj) A ds = JU (V • A) dv (JJ> ipds — HI Wipdv ’ S V U> (n x A) ds = HI (V x A) dv S V h-II (V x A) • ds c s j> vdl = J J (n x V p) ds c ' s 111 ^^ ‘ ^v) dv = $ (V^ ■ n) ds V s II! ~ dv ~ u • n] ds—u jj>T) dA = Q,(V) B ■ dA = 0 E.dl=-^ ds W H-dl = //,s+-f as at VB =0 V x E= - ffj Hi JJdv E - dA <9B dt Q(V) £o B • dA = 0 d$B,s 9v E • dl dt 0E t r m ££ g V x B = ¿/0J + n^ B • dl = f.t0Is + ^o-o ' , dip dip ds
гуманитариям не понять,магия света
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