How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$?

Question

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$?

for example, $u= x+2y, v=xe^y$, how to find the inverse?

Answer 1

A general advice would be to try and express one of $x$ and $y$ by manipulating $u$ and $v$. It is hard to be more specific, because the niceness of $f$ (e.g. its being expressible in terms of elementary functions) does not generally imply niceness of $f^{-1}$.

For example, in the present case, we can isolate an expression in terms of $x$ by eliminating the $e^y$ part of $v$:

$$v\,\exp \left(-\frac u2\right) = x\exp y\exp\left(-\frac x2-y\right) = x \exp\left(-\frac x2\right)$$

From here on, we find that, using the Lambert W-function:

\begin{align} v \exp\left(-\frac u2\right) &= x \exp\left(-\frac x2\right)\\ -\frac v2\exp\left(-\frac u2\right) &= -\frac x2 \exp\left(-\frac x2\right)\\ W\left(-\frac v2\exp\left(-\frac u2\right)\right) &= -\frac x2\\ -2 W\left(-\frac v2\exp\left(-\frac u2\right)\right) &= x \end{align}

and using this expression for $x$, we can use $y = \frac12(u-x)$ to find $y$ in terms of $u$ and $v$.