Injectivity of $g(x,y)=(2ye^{2x},xe^y)$ from $(0,1)$, onto $(2,0)$

Question

I want to prove the injectivity of the function in the problem below given the restrictions. This question is from Munkres' "Analysis on Manifolds" in the chapter on the Inverse Function theorem. The problems before it ask the same question but with different restrictions. In those cases I had to either convert to polar form, change to complex coordinates, to prove injectivity, or just leave the function in its original form and solve from there.

I looked for ways to convert to complex coordinates but didn't see any map to do this with, and when left in original form, I reduced to

$$y_1e^{2x_1}=y_2e^{2x_2} \text{ for the x coordinate}$$

$$x_1e^{y_1}=x_2e^{y_2} \text{ for the y coordinate}$$

Not sure if bringing the restrictions in at this point helps, or if this is the wrong way to proceed.

Thanks!

Answer 1

I think the point is to use the inverse function theorem, given that the exercise is at the end of that section. The Jacobian determinant is $2e^{2x +y}(2xy-1)$, which evaluates to $-2e$ at $(0,1)$. Hence the conclusion follows.