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The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p0711701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p0711702.png" /> related to rectangular Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p0711703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p0711704.png" /> by the formulas
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$#A+1 = 16 n = 0
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$#C+1 = 16 : ~/encyclopedia/old_files/data/P071/P.0701170 Parabolic coordinates
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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/p/p071/p071170/p0711705.png" /></td> </tr></table>
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{{TEX|auto}}
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p0711706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p0711707.png" />. The coordinate lines are two systems of mutually orthogonal parabolas with oppositely-directed axes.
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The numbers  $  u $
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and  $  v $
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related to rectangular Cartesian coordinates  $  x $
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and  $  y $
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by the formulas
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 +
$$
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x  =  u  ^ {2} - v  ^ {2} ,\  y  =  2uv ,
 +
$$
 +
 
 +
where $  - \infty < u < \infty $
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and  $  0 \leq  v < \infty $.  
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The coordinate lines are two systems of mutually orthogonal parabolas with oppositely-directed axes.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071170a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071170a.gif" />
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The Lamé coefficients (or scale factors) are given by:
 
The Lamé coefficients (or scale factors) are given by:
  
<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/p/p071/p071170/p0711708.png" /></td> </tr></table>
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$$
 +
L _ {u}  = L _ {v}  = 2 \sqrt {u  ^ {2} + v  ^ {2} } .
 +
$$
  
 
The area element is given by:
 
The area element is given by:
  
<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/p/p071/p071170/p0711709.png" /></td> </tr></table>
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$$
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d \sigma  = 4( u  ^ {2} + v  ^ {2} )  du  dv .
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$$
  
 
The fundamental operators of vector analysis are given by:
 
The fundamental operators of vector analysis are given by:
  
<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/p/p071/p071170/p07117010.png" /></td> </tr></table>
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$$
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\mathop{\rm grad} _ {u}  f  = \
  
<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/p/p071/p071170/p07117011.png" /></td> </tr></table>
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\frac{1}{2 \sqrt {u  ^ {2} + v  ^ {2} } }
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\frac{\partial  f }{\partial  u }
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,
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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/p/p071/p071170/p07117012.png" /></td> </tr></table>
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$$
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\mathop{\rm grad} _ {v}  f  =
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\frac{1}{2 \sqrt {u  ^ {2}
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+ v  ^ {2} } }
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\frac{\partial  f }{\partial  v }
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,
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$$
  
<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/p/p071/p071170/p07117013.png" /></td> </tr></table>
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$$
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\mathop{\rm div}  a  =
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\frac{1}{2 \sqrt {u  ^ {2} + v  ^ {2} } }
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\left (
 +
\frac{\partial  a _ {u} }{\partial  u }
  
In parabolic coordinates the [[Laplace equation|Laplace equation]] allows separation of variables.
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+
 +
\frac{\partial  a _ {v} }{\partial  v }
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\right ) +
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\frac{ua _ {u} + va _ {v} }{2 \sqrt {( u  ^ {2} + v  ^ {2} )  ^ {3} } }
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,
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$$
  
 +
$$
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\Delta f  = 
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\frac{1}{4( u  ^ {2} + v  ^ {2} ) }
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\left (
 +
\frac{\partial  ^ {2} f }{
 +
\partial  u  ^ {2} }
 +
+
 +
\frac{\partial  ^ {2} f }{\partial  v  ^ {2} }
 +
\right ) .
 +
$$
  
 +
In parabolic coordinates the [[Laplace equation|Laplace equation]] allows separation of variables.
  
 
====Comments====
 
====Comments====
Using complex functions the coordinate transformation can be described by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p07117014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p07117015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071170/p07117016.png" />.
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Using complex functions the coordinate transformation can be described by $  \widetilde{z}  = z  ^ {2} $,  
 +
where $  z= u+ iv $
 +
and $  \widetilde{z}  = x+ iy $.
  
 
For parabolic coordinates in space see [[#References|[a1]]].
 
For parabolic coordinates in space see [[#References|[a1]]].

Latest revision as of 08:05, 6 June 2020


The numbers $ u $ and $ v $ related to rectangular Cartesian coordinates $ x $ and $ y $ by the formulas

$$ x = u ^ {2} - v ^ {2} ,\ y = 2uv , $$

where $ - \infty < u < \infty $ and $ 0 \leq v < \infty $. The coordinate lines are two systems of mutually orthogonal parabolas with oppositely-directed axes.

Figure: p071170a

The Lamé coefficients (or scale factors) are given by:

$$ L _ {u} = L _ {v} = 2 \sqrt {u ^ {2} + v ^ {2} } . $$

The area element is given by:

$$ d \sigma = 4( u ^ {2} + v ^ {2} ) du dv . $$

The fundamental operators of vector analysis are given by:

$$ \mathop{\rm grad} _ {u} f = \ \frac{1}{2 \sqrt {u ^ {2} + v ^ {2} } } \frac{\partial f }{\partial u } , $$

$$ \mathop{\rm grad} _ {v} f = \frac{1}{2 \sqrt {u ^ {2} + v ^ {2} } } \frac{\partial f }{\partial v } , $$

$$ \mathop{\rm div} a = \frac{1}{2 \sqrt {u ^ {2} + v ^ {2} } } \left ( \frac{\partial a _ {u} }{\partial u } + \frac{\partial a _ {v} }{\partial v } \right ) + \frac{ua _ {u} + va _ {v} }{2 \sqrt {( u ^ {2} + v ^ {2} ) ^ {3} } } , $$

$$ \Delta f = \frac{1}{4( u ^ {2} + v ^ {2} ) } \left ( \frac{\partial ^ {2} f }{ \partial u ^ {2} } + \frac{\partial ^ {2} f }{\partial v ^ {2} } \right ) . $$

In parabolic coordinates the Laplace equation allows separation of variables.

Comments

Using complex functions the coordinate transformation can be described by $ \widetilde{z} = z ^ {2} $, where $ z= u+ iv $ and $ \widetilde{z} = x+ iy $.

For parabolic coordinates in space see [a1].

References

[a1] R. Sauer (ed.) I. Szabó (ed.) , Mathematische Hilfsmittel des Ingenieurs , 1 , Springer (1967) pp. 98
[a2] L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1960) pp. 151ff (Translated from Russian)
How to Cite This Entry:
Parabolic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_coordinates&oldid=48105
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article