TITLE: Using paraboloidal coordinates
QUESTION [5 upvotes]: I have the 3-dimensional paraboloidal coordinates
$$s_{\pm}=\sqrt{x^2+y^2+z^2}\pm z$$
$$\phi=ArcTan(y/x)$$
with the inverse transformation
$$x=\sqrt{s_+ \cdot s_-}\cdot cos(\phi)$$
$$y=\sqrt{s_+ \cdot s_-}\cdot sin(\phi)$$
$$z=\frac{s_+ - s_-}{2}$$
Questions:

Properties in integration: Suppose I have $ I= \int_{0}^{\infty} ds_+ \int_{0}^{\infty} ds_- f(s_+,s_-) \cdot \delta(s_+ - t)$, can I naively apply the Dirac-Delta and get $I=\int_{0}^{\infty} ds_- f(t,s_-)$ ?
Properties in differentiation: Suppose I have $D=\frac{\partial}{\partial s_-} e^{2\cdot s_-} \cdot s_+$, can I naively ignore the $s_+$ Term and get $D=2\cdot e^{2\cdot s_-} \cdot s_+$ ?

Every hint will be much appreciated!

REPLY [2 votes]: What you wrote is
$$
\begin{eqnarray}
s_+ &=& \sqrt{x^2+y^2+z^2} + z,\\
s_- &=& \sqrt{x^2+y^2+z^2} - z.\\
\end{eqnarray}
$$
As both are positive, we can take the square root, thus
$$
\begin{eqnarray}
t_+ &=& \sqrt{\sqrt{x^2+y^2+z^2} + z},\\
t_- &=& \sqrt{\sqrt{x^2+y^2+z^2} - z}.\\
\end{eqnarray}
$$
Note that
$$
\begin{eqnarray}
t_+ t_- &=& \sqrt{x^2+y^2},\\
\frac{t_+^2 - t_-^2}{2} &=& z,
\end{eqnarray}
$$
so
$$
\begin{eqnarray}
x &=& \cos(\phi) t_+ t_-,\\
y &=& \sin(\phi) t_+ t_-,\\
z &=& \frac{t_+^2 - t_-^2}{2}.
\end{eqnarray}
$$
These coordinates are known as Paraboloidal coordinates $(\phi,u,v)$.
What you have defined is $(\phi,s_+,s_-) = (\phi,u^2,v^2)$.
We can call then the partly squared paraboloidal or the NicoDean coordinates...