View source for Paraboloidal coordinates

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Numbers  $  u $, 
$  v $
and  $  w $
related to the rectangular Cartesian coordinates  $  x $, 
$  y $
and  $  z $
by the formulas

$$ 
x  =  2 uw  \cos  v,\ \ 
y  =  2uw  \sin  v,\ \ 
z  =  u  ^ {2} - w  ^ {2} ,
$$

where  $  0 \leq  u < \infty $, 
$  0 \leq  v < 2 \pi $, 
$  0 \leq  w < \infty $. 
The coordinate surfaces are two systems of paraboloids of revolution with oppositely-directed axes ( $  u = \textrm{ const } $
and  $  w = \textrm{ const } $) 
and half-planes ( $  v = \textrm{ const } $). 
The system of paraboloidal coordinates is orthogonal.

The Lamé coefficients (or scale factors) are

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

The element of surface area is

$$ 
d \sigma =
$$

$$ 
= \ 
4 \sqrt {( u  ^ {2} + w  ^ {2} ) u  ^ {2} w  ^ {2} ( du  ^ {2}
+ dw  ^ {2} )  dv  ^ {2} + ( u  ^ {2} + w  ^ {2} )( du  dw)  ^ {2} } .
$$

The volume element is

$$ 
dV  =  8( u  ^ {2} + w  ^ {2} ) uw  du  dv  dw.
$$

The fundamental operations of vector analysis are

$$ 
 \mathop{\rm grad} _ {u}  \phi  = \ 

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

\frac{1}{2uw}
 
\frac{\partial  \phi }{\partial  v }
 ,
$$

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

$$ 
 \mathop{\rm div}  \mathbf a  =  
\frac{1}{2uw \sqrt {( u  ^ {2} + w  ^ {2} )  ^ {3} } }
 \times
$$

$$ 
\times 
[ w \mathbf a _ {u} ( 2u  ^ {2} + w  ^ {2} ) + u \mathbf a _ {w} ( u  ^ {2} + 2w  ^ {2} )] +
$$

$$ 
+ 

\frac{1}{2 \sqrt {u  ^ {2} + w  ^ {2} } }
 \left ( 
\frac{\partial  \mathbf a _ {u} }{\partial 
u }
 + 
\frac{\partial  \mathbf a _ {w} }{\partial  w }
 \right )
+ 
\frac{1}{2uw}
 
\frac{\partial  \mathbf a _ {v} }{\partial  v }
 ;
$$

$$ 
 \mathop{\rm rot} _ {u}  \mathbf a  =  
\frac{1}{2uw}
 
\frac{\partial  \mathbf a _ {w} }{\partial  v }
 - 
\frac{1}{2w \sqrt {u  ^ {2} + w  ^ {2} } }
 \left (
\mathbf a _ {v} + w 
\frac{\partial  \mathbf a _ {v} }{\partial  w }
 \right ) ,
$$

$$ 
 \mathop{\rm rot} _ {v}  \mathbf a  =  
\frac{1}{2 \sqrt {( u  ^ {2} + w  ^ {2} )  ^ {3} } }
 ( w \mathbf a _ {u} - \mathbf a _ {w} ) +
$$

$$ 
+ 

\frac{1}{2 \sqrt {u  ^ {2} + w  ^ {2} } }
 \left ( 
\frac{\partial  \mathbf a _ {u}  }{\partial  w }
 - 
\frac{\partial  \mathbf a _ {w} }{\partial  u }
 \right ) ;
$$

$$ 
 \mathop{\rm rot} _ {w}  \mathbf a  =  
\frac{1}{2w( u  ^ {2} + w  ^ {2} ) }

\left ( \mathbf a _ {v} + u 
\frac{\partial  \mathbf a _ {v} }{\partial 
u }
 \right ) - 
\frac{1}{2uv}
 
\frac{\partial  \mathbf a _ {u} }{\partial  v }
 ;
$$

$$ 
\Delta \phi  =  
\frac{1}{4( u  ^ {2} + w  ^ {2} ) }
 \left
[ 
\frac{\partial   ^ {2} \phi }{\partial  u  ^ {2} }

+ 
\frac{1}{u}
 
\frac{\partial  \phi }{\partial  u }
\right . +
$$

$$ 
+ \left .
\left ( 
\frac{1}{u  ^ {2} }
 + 
\frac{1}{w  ^ {2}
 }
 \right ) 
\frac{\partial   ^ {2} \phi }{\partial  v  ^ {2} }
 + 
\frac{\partial 
 ^ {2} \phi }{\partial  w  ^ {2} }
 + 
\frac{1}{w}
 
\frac{\partial  \phi }{\partial  w }
 \right ] .
$$

====Comments====
These coordinates are also called rotation parabolic coordinates.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Sauer (ed.)  I. Szabó (ed.) , ''Mathematische Hilfsmittel des Ingenieurs'' , '''1''' , Springer  (1967)  pp. 96</TD></TR></table>