Paraboloidal coordinates

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

[a1]	R. Sauer (ed.) I. Szabó (ed.) , Mathematische Hilfsmittel des Ingenieurs , 1 , Springer (1967) pp. 96