I'm working on a differential geometry question using flows, and I'm trying to find the inverse of a specific curve (specifically a curve at a regular point).
The function is: $\psi(t,s)=(\frac{1}{2}s(e^{t}+e^{-t}),\frac{1}{2}s(e^{t}-e^{-t}))$, and I want the inverse to be of the form $\psi^{-1}(x,y)=(f(x,y),g(x,y))$ (inverse meaning composition both ways reduces to the identity i.e. $\frac{1}{2}g(x,y)(e^{f(x,y)}+e^{-f(x,y)})=x$ and the same for $y$).
Any ideas for the inverse? The fact that $t$ and $s$ aren't separable here is what's causing me so much trouble, and I think some clever usage of the $ln$ function or something similar could help me work it out, but so far I'm having no luck. Is there a general way to approach these types of problems? Any help would be appreciated!