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Difference between revisions of "Bicylindrical coordinates"

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The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161202.png" /> related to the rectangular Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161204.png" /> by the formulas
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161205.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161206.png" />. The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161207.png" />), the family of circular cylinders orthogonal to the former (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161208.png" />), and the planes (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b0161209.png" />). The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b01612010.png" />-plane parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b01612011.png" />-axis.
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The numbers  $  \tau , \sigma $
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and  $  z $
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related to the rectangular Cartesian coordinates  $  x, y $
 +
and  $  z $
 +
by the formulas
 +
 
 +
$$
 +
=
 +
\frac{a  \sinh  \tau }{\cosh  \tau - \cos  \sigma }
 +
,\ \
 +
=
 +
\frac{a  \sin  \sigma }{\cosh  \tau - \cos  \sigma }
 +
,\ \
 +
z  =  z,
 +
$$
 +
 
 +
where  $  0 \leq  \sigma < \pi , -\infty < \tau < \infty $.  
 +
The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes ( $  \tau = \textrm{ const } $),  
 +
the family of circular cylinders orthogonal to the former ( $  \sigma = \textrm{ const } $),  
 +
and the planes ( $  z = \textrm{ const } $).  
 +
The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the $  xy $-
 +
plane parallel to the $  z $-
 +
axis.
  
 
The Lamé coefficients (cf. [[Lamé coefficients|Lamé coefficients]]) are:
 
The Lamé coefficients (cf. [[Lamé coefficients|Lamé coefficients]]) are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b01612012.png" /></td> </tr></table>
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$$
 +
L _  \sigma  = L _  \tau  = \
 +
 
 +
\frac{a  ^ {2} }{( \cosh  \tau - \cos  \sigma )  ^ {2} }
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,\ \
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L _ {z}  = 1.
 +
$$
  
 
The Laplace operator is:
 
The Laplace operator is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016120/b01612013.png" /></td> </tr></table>
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$$
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\Delta f  =
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\frac{1}{a  ^ {2} }
 +
 
 +
( \cosh  \tau - \cos  \sigma )  ^ {2}
 +
\left (
 +
 
 +
\frac{\partial  ^ {2} f }{\partial  \sigma  ^ {2} }
 +
+
 +
 
 +
\frac{\partial  ^ {2} f }{\partial  \tau  ^ {2} }
 +
\
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\right ) +
 +
 
 +
\frac{\partial  ^ {2} f }{\partial  z  ^ {2} }
 +
.
 +
$$

Latest revision as of 10:59, 29 May 2020


The numbers $ \tau , \sigma $ and $ z $ related to the rectangular Cartesian coordinates $ x, y $ and $ z $ by the formulas

$$ x = \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } ,\ \ y = \frac{a \sin \sigma }{\cosh \tau - \cos \sigma } ,\ \ z = z, $$

where $ 0 \leq \sigma < \pi , -\infty < \tau < \infty $. The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes ( $ \tau = \textrm{ const } $), the family of circular cylinders orthogonal to the former ( $ \sigma = \textrm{ const } $), and the planes ( $ z = \textrm{ const } $). The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the $ xy $- plane parallel to the $ z $- axis.

The Lamé coefficients (cf. Lamé coefficients) are:

$$ L _ \sigma = L _ \tau = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } ,\ \ L _ {z} = 1. $$

The Laplace operator is:

$$ \Delta f = \frac{1}{a ^ {2} } ( \cosh \tau - \cos \sigma ) ^ {2} \left ( \frac{\partial ^ {2} f }{\partial \sigma ^ {2} } + \frac{\partial ^ {2} f }{\partial \tau ^ {2} } \ \right ) + \frac{\partial ^ {2} f }{\partial z ^ {2} } . $$

How to Cite This Entry:
Bicylindrical coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicylindrical_coordinates&oldid=46051
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article