I'm studying Multivariate Calculus and I've just studied the inverse theorem and now I'm doing some exercises.
There are some questions about local injectivity that are causing some doubts. For example:
I have to study the local injectivity of this function $f(x,y) = (x^2 + 2xy + y^2,x+y)$, we can conclude that f is a $C^1$ function also the Jacobian of $f$ is given by: $$ Jf_{(x,y)} = \begin{bmatrix} 2x + 2y & 2x + 2y \\ 1 & 1\\ \end{bmatrix} $$ then $det(Jf_{(x,y)})=0$ for all $(x,y) \in \mathbb{R^2}$
Therefore, I can't apply the inverse theorem for any point, then It's not possible to conclude local injectivity for any point.
So I've decided to study injectivity of $f(x,y) = ((x+y)^2,x+y)$, and here I conclude that $f$ is not injective because $f(a,b) = f(b,a)$.
I'd like to know, is there any gap in my proof? Is there a easier to do same thing?
Thank you!
You are right that $f$ is not globally injective, since $f(a,b) = f(b,a)$, but if you want to prove that $f$ is not locally injective you should consider a small neighborhood around a specific point. In other words, can you prove that $f(a+\epsilon, b-\epsilon) = f(a-\epsilon, b+\epsilon)$ for small $\epsilon$?