Parabolic coordinates

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)