Lecture 3 Orthogonal Coordinate Systems .pdf

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1、Lecture3OrthogonalCoordinateSystemsReviewQ-3.1Whichofthefollowingproductsofvectordonotmakesense?Explain.rrrAgBCrrrACBg(√)rrrABC(√)rrABrAaArrrABCg(√)rrrrrrAggBCACBOutline•Generalizedorthogonalcurvilinearcoordinates•ThreesetsoforthogonalCoordinates•IntegralsContainingVec

2、torFunctions3.1GeneralorthogonalcurvilinearcoordinatesFromamathematicalpointofviewitisveryconvenienttoworkwithvectorswhentheyareresolvedintocomponentsalongthreemutuallyorthogonal(perpendicular)directions.3.1.1BaseVectors(unitvector):aˆu1aˆu2aˆu3Inageneralright-handed,orthogonal,curvi

3、linearcoordinatesystemthefollowingrelationsaresatisfied:aˆuaauu123aˆgaaggaaa0u1u2u2u3uu31aˆuaauu231aˆgaaggaaa1u1u1u2u2uu33aˆuaauu312rrAaˆAaaAA222u1u1u2u2uu33AAAuAAuu123rrrrrre.g.GiventhreevectorsA,BCand,obtaintheexpressionsofCgAB,intheorthogonalcurvilinearcoor

4、dinatesystemu1,,uu23rrrSolutionA,BCandcanbewrittenintheorthogonalcoordinatesu1,,uu23rrrAau1Au1au2Au2au3Au3,,Bau1Bu1au2Bu2au3Bu3Cau1Cu1aau2CCu2uu33rrrFABauAuBuAuBuAuaauAuBuAuBuuAuBuABuu1232312311331221aaauuu123=AAAuuu123BBBuuu123rrCgFCFCFCFuuuuuu112233=Cu

5、AuBuAuBuAuCuAuBuAuBuCuAuBuABuu1232312311331221CCCuuu123=AAAuuu123BBBuuu1233.1.2differentialelementInvectorcalculusline,surface,andvolumeintegralsarealwaysperformed.Ineachcaseweneedtoexpressthedifferentiallength-changecorrespondingtoadifferentialchangeinoneofthecoordinates.

6、Ødifferentiallengthhdlihiidui1,2,or3i:Metriccoefficient(Lamécoefficients)Adirecteddifferentiallength-changeinanarbitrarydirectioncanbewrittenasthevectorsumofthecomponentlength-changes:rrdlaudl1aauudl23dlordlauh1du1aauuh2du2h33du123123Ødifferentialvolumedvhhhdududu123

7、123Ødifferentialareardsadsnr电流面密度矢ds1audl23dlds1h2h3du23du量1aaSimilarly,thedifferentialareasnormaltounitvectorsu2andu3arerespectively,dshhdudu21313anddshhdudu312123.2Cartesian(orRectangular)Coordinatesu1,u23,ux,,yzaaaxyzaaayzxaaazxyuuurPositionvectorOPaxx1aayzyz11r

8、AaAaaAArv