# 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.

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