cylindrical coordinate
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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 ).