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Cylindrical Coordinate

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposinga height ( ) axis. Unfortunately, there are a number of different notations used for the other two coordinates. Either

or is used to refer to the radial coordinate and either or to the azimuthal coordinates. rf!en ("#$%), for

instance, uses , while &eyer ("#$') uses . n this wor!, thenotation  is used.

he following table summarizes notational con*entions used by a number of authors.

(radial, azimuthal,*ertical)

reference

this wor!, &eyer ("#$', p. +"+)

(Rr, Ttheta, Zz)   SetCoordinatesCylindrical in

the Mathematica pac!age VectorAnalysis`

 rf!en ("#$%, p. #%)

oon and /pencer ("#$$, p. "+)

0orn and 0orn ("#1$, p. 12)

orse and 3eshbach ("#%4)

n terms of the Cartesian coordinates ,

(")

(+)

(4)

where , , , and the in*erse tangent must be suitably defined to ta!e the correct

5uadrant of into account.

n terms of , , and

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(6)

(%)

(1)

7ote that orse and 3eshbach ("#%4) define the cylindrical coordinates by

(')($)

(#)

where and .

he metric elements of the cylindrical coordinates are

("2)

("")

("+)

so the scale factors are

("4)

("6)

("%)

he line element is

(16)

and the *olume element is

(17)

he 8acobian is

(18)

  Cartesian *ector  is gi*en in cylindrical coordinates by

(19)

o find the unit *ectors,

(+2)

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(+")

(++)

9eri*ati*es of unit *ectors with respect to the coordinates are

(+4)

(+6)

(+%)

(+1)

(+')

(+$)

(+#)

(42)

(4")

he gradient operator in cylindrical coordinates is gi*en by

(32)

so the gradient components become

(44)

(46)

(4%)

(41)

(4')

(4$)

(4#)

(62)

(6")

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he Christoffel symbols of the second !ind in the definition of isner et al. ("#'4, p. +2#) are gi*en by

(6+)

(64)

(66)

he Christoffel symbols of the second !ind in the definition of rf!en ("#$%) are gi*en by

(6%)

(61)

(6')

(:alton "#1'; rf!en "#$%, p. "16, Ex. 4.$."2; oon and /pencer "#$$, p. "+a).

he co*ariant deri*ati*es are then gi*en by

(48)

are

(6#)

(%2)

(%")

(%+)

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(%4)

(%6)

(%%)

(%1)

(%')

Cross products of the coordinate axes are

(%$)

(%#)

(12)

he commutation coefficients are gi*en by

(61)

&ut

(62)

so , where . lso

(63)

so , . 3inally,

(64)

/ummarizing,

(1%)

(11)

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(1')

ime deri*ati*es of the *ector  are

(1$)

(1#)

('2)

('")

('+)

/peed is gi*en by

('4)

('6)

ime deri*ati*es of the unit *ectors are

('%)

('1)

('')

('$)

('#)

($2)

he con*ecti*e deri*ati*e is

($")

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($+)

o rewrite this, use the identity

(83)

and set , to obtain

(84)

so

(85)

hen

($1)

($')

he curl in the abo*e expression gi*es

($$)

($#)

so

(#2)

(#")

(#+)

:e expect the gradient term to *anish since speed does not depend on position. Chec! this using the

identity ,

(#4)

(#6)

Examining this term by term,

(#%)

(#1)

(#')

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(#$)

(##)

("22)

("2")

("2+)

("24)

("26)

("2%)

("21)

("2')

so, as expected,

(108)

:e ha*e already computed , so combining all three pieces gi*es

("2#)

(""2)

(""")

he di*ergence is

(""+)

(""4)

(""6)

(""%)

or, in *ector  notation

(116)

he curl is

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(""')

he scalar  <aplacian is

(""$)

(""#)

he *ector <aplacian is

(120)

he =elmholtz differential e5uation is separable in cylindrical coordinates and has /t>c!el determinant  (for ,

, ) or (for orse and 3eshbach?s , , and ).