Proving that $2$-D parabolic coordinates are orthogonal

Question

How can we prove that the parabolic coordinate system in two dimensions is orthogonal? I tried using the dot product, but don't know where to start or what basis vectors can be used in two dimensions. In three dimensions I would normally show that the dot product of the basis vectors is zero, but don't know the $2$-D basis vectors for parabolic coordinates, any hints?

The $2$-D system is defined as:

$$ x= \mu \kappa$$ $$ y = \frac{1}{2}(\mu^2-\kappa^2) $$

Answer 1

You can think of your coordinates as a parameterization ${\bf r}$ of the appropriate subset of the $xy$-plane in the coordinate. Then, in the coordinate basis $(x, y)$, \begin{align} \partial_{\kappa} &= {\bf r}_{\kappa} = (\mu, -\kappa) \\ \partial_{\mu} &= {\bf r}_{\mu} = (\kappa, \mu) . \end{align}

Computing gives $$\partial_{\kappa} \cdot \partial_{\mu} = 0.$$