COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS
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1 Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate sstem tat uses an constant o smmet n te poblem. l O Constant: l constant moton n a plane onl vaable: Pont 0 s fed. Ts poblem could appea complcated n an X-Y coodnate sstem. Te poblem s smple n te coodnate sstem wt l constant. Z O E Β P We want te electc feld E and te magnetc nducton B due to a cuent along a conductng we as sown. Ts poblem as an aal smmet. Instead of te usual coodnate sstem let us use as sown. t pont P B µ 0 π o P E Smlal te coodnate sstem s best fo fndng te electc feld at pont P due to cage at pont O. E. Specal smmet caactees ts poblem. II. GENERLIZED COORDINTE SYSTEMS & COORDINTE TRNSFORMTION Genealed Coodnate Sstems Let be a egula coodnate sstem wee eac epesents tance. Let be te coodnate sstem to wc we want to cange and wee ae NOT necessal tances. coodnate sstem wee te... n do not necessal epesent tances s a genealed coodnate sstem. coodnate sstem s sad to be cuvlnea wen te
2 Dola Bagaoo 0 coodnate sufaces ae not necessal planes. In a genealed coodnate sstem d does not geneall epesent tance-element le d d and d do n te tee dmensonal -D Catesan coodnate sstem. Coodnate Tansfomatons Ke : Ts cange of coodnate sstem must be done wtout cangng te pscal poblem. Pactcall ts means tat te tansfomaton must conseve "tance." tansfomaton fom coodnates to coodnates follows te dagam below: tansfom sometc... E. Te specfc sometc so same metc tance tansfomaton o tansfom utled n coodnate tansfomatons s an otogonal tansfom.e. one wose mat s otogonal. Let be te unt vectos n a ectangula Catesan coodnate sstem. nd let be te unt vectos n a genealed coodnate sstem nown as te specal pola coodnate sstem. Ten te above tansfomaton dagam concetel lea to:. *0 sn sn sn sn sn sn tansfom te Mat of 0 sn sn sn sn sn sn E. s otogonal T so o Ts eample llustates te above dagam fo coodnate tansfoms. loo at tese euatons lea to te mat fom below.
3 Dola Bagaoo 0 Fundamental Popetes of Genealed Coodnate Sstems Fom E. f ten d d d d and smlal fo and. Wt tenso notaton we ave d d d d d d wt summaton ove a epeated nde. Ten d d d o d dd wt summaton ove bot ndces and. Note ve well te fact tat n multplng d b tself we canged te dumm nde fom one epesson of d.e. to anote d. Te suae of te placement element s n -D d d d Coodnate tansfomatons conseve te suae of te dffeental placement element. In -D d d d [ ] dd d d 44 Sum ove Wt Te mat fomed b te s called te metc tenso. III. ORTHOGONL COORDINTE SYSTEMS coodnate sstem s otogonal f te metc tenso s dagonal: δ N Fo otogonal sstems te N ae eo ecept fo. Te N ae numbes of functons. Fo otogonal coodnate sstems d d d. One obtans te followng esult b applng te followng fundamental popet: In a genealed and otogonal coodnate sstem d d and d ave te dmenson of tance ave te dmenson of suface; and s te volume element. Moe on Otogonal Coodnate Stems: Consult te tetboo and ote boos about te eleven well nown genealed and otogonal coodnate sstems etensvel utled n scence engneeng and matematcs. Gven tat αδ ten onl and ae non-eo n a -D otogonal coodnate
4 Dola Bagaoo 0 sstem. So we note as as and as wtout wong about ambgut. I. PPLICTIONS: Ke dffeental opeatos ae gven below n tems of genealed coodnate vaables fo otogonal coodnate sstems. s d Dffeentaton Rule Gadent. Te ae unt vectos. [ ] E E E E Dvegence [ ] } { } { } { Laplacan of. Laplacan Opeato * b b b B Cul Opeaton 4 n - Dmenson : element olume : Suface element Dsplacement elements: d d d d d d d σ
5 Dola Bagaoo 0. PPLICTIONS: EXMPLES OF ORTHOGONL COORDINTE SYSTEMS. Rectangula Catesan Coodnate Sstem Z Y Metc tenso: Otogonalt means tat Unt vectos... X Once te metc tenso s nown we can wte down te placement elements: d. d d d σ. dd dd suface elements: d dd etc. Te volume element dv ddd. s s.. E.. E... E E E E E [ ] Ote opeatos can smlal be woed out.. Ccula Clndcal Coodnate Sstem ẑ ρ ρ ρ Ranges ρ : 0 : 0 π Z : ρ ρ ρ sn Z Z Note well tat ρ.. ρ. 0 dρ ρd d dv dρ ρd d ρ dρ ρ d Homewo: Wte down Mawell's electomagnetc euatons and te defnton of te electc feld and of te magnetc nducton n te ccula clndcal coodnate sstem. 5
6 Dola Bagaoo 0. Specal Pola Coodnate sstem X Z B Y Z OBsn X OB sn Y OBsn sn sn sn Ranges of coodnate vaables : : 0 : 0 π : 0 π 0 0 Wt te above nfomaton te metc tenso sn. d d We ave d d d snd So d σ d d d snd d snd sndd dv snddd Te volume element Homewo: Repeat fo te specal pola coodnate sstem te assgnment gven fo te ccula clndcal coodnate sstem. 6
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