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    5

DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5

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Cylindrical Coordinates

Divergence

1 d / x                                              1 OAs dAz

V • A =--------------------- (rAr) +-------------- ^ + ------- -

r dr                                                  r d(f) dz

Gradient

,                    x         d/          z         x          Id/             z         x          df

(V/ r =                                  (V/ * = --5-; (V/)z = /

ar                                                          r ocp                             Oz

Curl

ZV7 . \ 1 dAz dA( (V x A)r = -

r d(j) dz

dAr dAz

(V X :

dz dr

Id, x 1 dAr VxA2 = --r^---------------------- -f

r ar                                                      r ocp

Laplacian

r dr V dr / r2 dcf)2 dz2

Laplacian of a vector

(V2A)r = V2Ar-4^-^:

rz ocp rz

o .                                 2              2 dAr A(

(V2A)0 = V2a0 + —

r2 <9(/> r'

(V2A)z = V2AZ

Components of (A • V)B

dBr A&BC

(A • VB)r = Ar-------------- + —--------- L + A

A(f>

dBr

r

dcj)

A^

 

r

d</>

A(f>

dBz

z dz

(A . VB), =                          +                                           + ^

Oz                          r

(A • VB)2                            +                     +

ar r ocp                                                                              oz

Divergence of a tensor

Tx 1 d /                                                     1 dTsr dTzr T4

(V • T)r = - — (rTrr) + -—^ +                                                            '

r <9r                                                        r <9(/>            dz              r

/Y7 "7~\                      1 d z ^ x , 1                             , dTz<f> ,

(V • I )«£ = - — (rTr0) H------------------------- —— + —--------- 1------

r ar                                                           r ocp               oz             r

1 d z                                                         1                  dTzz

(V • = - -(rTr,) + --^ + ^— r ar                 r ocp               oz

Spherical Coordinates

Divergence

1 a 2 . . 1 d............................................................................ 1 OA,

V • A = --(r Ar) + -———-(sin<9A0) +

r2 dr                                                      r sin 0 86                           r sin 0 dcf)

Gradient

,                   x         d/         z          x          1 df          z         x               1      df

(V/ r =                                (V/), =                          (V/ * = —

ar                                                          r at/                              r sin tf ocp

Curl

1 d z                                                                               x 1 dA,

(V X A)r = —:-------------------------- (sin OA^)

r sin 6 d6                                                                 r sin 0 dcf)

1 dAr 1 d z (V x A), = —--f - --(r^) r sin 6/ ocp r Or

1 d z x 1 dAr

(VxA)^--(^)-----------------------------------------

r ar                                                            r at/

Laplacian

V2/- 1 d fr2d^) + 1 ° (s'n6df\ + 1 ^ r2 dr \ dr J r2 sin 6 d6 \ d6 J r2 sin2 6 dcf)2

Laplacian of a vector

(V2A)r = V2Ar -

2Ar 2 dAe 2 cot 0Ae                                  2 dA(

r r2 <9(9                            r2                r2 sin 0 dcj)

2                                                2              2 dAr                 Ae                2 cos 9 dA4

(V A)fl = V A0 + — —------------------------------ 2-2/3----------- 2-2/3 ~aT

r^ atx r^ sir 6/ rz sir 6 ocp

z 2 x                                          2                   A&                    2 dAr 2 cos 6> dAe

r2 sin2 0 r2 sin 0 dcj) r2 sin2 0 dcj)

Components of (A • V)B

ZA                                    , dBr Ae dBr , A& dBr AeBe + A^BC

(A • \/t>)r — Ar ——— +---------------------- —— +

dr r d6 r sin 0 dcj)                                                                                               r

/A v^x A dB° ■ Ae dBe ^ A^ dBe , AeBr cot OA^B, (A-VB )e = Ar—           1          ——I   r^^— +

dr r d6 r sin 0 d(j)                                                                           r                       r

/a v^x a dB<t> , Ae dB<t> , A<t> dB<t> , A4>Br , cot OA^Bi

(A • V-dJ^ = Ar------------- +--------------- +--------------------- + ------------  + -----------------

dr r dO r sin 0 d(j)                                                                            r                      r

Divergence of a tensor

Tx                                  1 d , o                          1       d /

(V- T)r = —— (r2Trr) + —- —(sin0T9r) r^ ar  r sin u otf

1                            Tee + T4

r sin 0 dcj)

1 d . 2r_ . 1 d

(V- T)e = —— (r^Tr0) + —- — (sin^T^)

r^ ar                                                  r sin tx at/

1 dT^e Ter cot

r sin 0 <9(/>            r

1 d , 2,_ . 1 d

(V- T)^ = — — (7-^) + —- — (sinflT^) r^ ar            r sm 6/ at/

1 dT^^                                  cot OTr,

r sin 0 dcj)               r                  r

Авторы: 1379 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 1908 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я