Review of Coordinate Systems

Transcription

1 Review o Coodinate Sstems good undestanding o coodinate sstems can be ve helpul in solving poblems elated to Mawell s Equations. The thee most common coodinate sstems ae ectangula (,, ), clindical (,, ), and spheical (,, ). Unit vectos in ectangula, clindical, and spheical coodinates In ectangula coodinates a point P is speciied b,, and, whee these values ae all measued om the oigin (see igue at ight). vecto at the point P is speciied in tems o thee mutuall pependicula components with unit vectos i, j, and k (also called,, and ). The unit vectos i, j, and k om a ighthanded set; that is, i ou push i intoj with ou ight hand, ou ight thumb will point along k diection. In clindical coodinates a point P is speciied b,,, whee is measued om the ais (o - plane) (see igue at ight). vecto at the point P is speciied in tems o thee mutuall pependicula components with unit vectos pependicula to the clinde o adius, pependicula to the plane though the ais at angle, and ẑ pependicula to the - plane at distance. The unit vectos,, ẑ om a ight-handed set.

2 In spheical coodinates a point P is speciied b,,, whee is measued om the oigin, is measued om the ais, and is measued om the ais (o - plane) (see igue at ight). With ais up, is sometimes called the enith angle and the aimuth angle. vecto at the point P is speciied in tems o thee mutuall pependicula components with unit vectos pependicula to the sphee o adius, pependicula to the cone o angle, and pependicula to the plane though the ais at angle. The unit vectos,, om a ighthanded set. Ininitesimal lengths and volumes n ininitesimal length in the ectangula sstem is given b d L d d d () and an ininitesimal volume b d v d d d () In the clindical sstem the coesponding quantities ae d L d d d (3) and d v d d d (4) In the spheical sstem we have d L d d d (5) and d v d d d (6)

3 Diection coes and coodinate-sstem tansomation s shown in the igue on the ight, the pojection o the scala distance on the ais is given b α whee α is the angle between and the ais. The pojection o on the ais is given b β, and the pojection on the ais b γ. Note that γ so γ. The quantities α, β, and γ ae called the diection coes. Fom the theoem o Pthagoas, α β γ (7) The scala distance o a spheical coodinate sstem tansoms into ectangula coodinate distance om which α (8) β (9) γ () α () β diection coes () γ (3) s the convese o (8), (9), and (), the spheical coodinate values (,, ) ma be epessed in tems o ectangula coodinate distances as ollows: (4) ( π ) (5) tan (6) Fom these and simila coodinate tansomations o spheical to ectangula and ectangula to spheical coodinates, we ma epess a vecto at some point P with spheical components,, as the ectangula components,, and, whee

4 (7) (8) (9) Note that the diection coes ae simpl the dot poducts o the spheical unit vecto with the ectangula unit vectos,, and : () β () γ () These and othe dot poduct combinations ae listed in the ollowing table: Rectangula Clindical Spheical Spheical Clindical Rectangula Σ ŷ ẑ ẑ ŷ ẑ ẑ Note that the unit vectos in the clindical and spheical sstems ae not the same. Fo eample, Spheical Clindical

5 In addition to ectangula, clindical, and spheical coodinate sstems, thee ae man othe sstems such as the elliptical, spheoidal (both polate and oblate), and paaboloidal sstems. lthough the numbe o possible sstems is ininite, all o them can be teated in tems o a genealied cuvilinea coodinate sstem. The undamental paametes o the ectangula, clindical, and spheical coodinate sstems ae summaied in the ollowing table: Coodinate sstem Rectangula Clindical Spheical Coodinates Range Unit Length Coodinate vectos elements suaces to o i d Plane constant to o j d Plane constant to o k d Plane constant to d Clinde constant to π d Plane constant to ẑ d Plane constant to d Sphee constant to π d Cone constant to π d Plane constant The ollowing two tables give the unit vecto dot poducts in ectangula coodinates o both ectangula-clindical and ectangula-spheical coodinates. Σ ŷ ẑ ẑ Rectangula-clindical poduct in ectangula coodinates Eample:

6 Rectangula-spheical poduct in ectangula coodinates Eample: Hee ae the tansomations o vecto components between coodinate sstems: Rectangula to clindical Clindical to ectangula Rectangula to spheical Spheical to ectangula ŷ ẑ

7 nd hee ae epessions o the gadient, divegence, and cul in all thee coodinate sstems: Rectangula coodinates Clindical coodinates Spheical coodinates ) ( ) (