Let $f : {\bf R}^2 \to {\bf R}^2$ and $f(x,y) = (x^2+2xy+y^2, 2x + 2y).$ Determine if $f$ is locally injective.

Question

Let $f : {\bf R}^2 \to {\bf R}^2$ and $f(x,y) = (x^2+2xy+y^2, 2x + 2y).$ Determine if $f$ is locally injective.

Noting first that $f$ is $C^1$ and computing the Jacobi one has that $$J_f(x,y) = \begin{bmatrix} 2x+2y && 2y+ 2x \\ 2 && 2\end{bmatrix}.$$

Thus the determinant $\det(J_f(x)) = 0$. However, this only means that the Jacobian matrix isn't invertible. The function might still be injective.

Any hints on how should I go about this and determine if the function is actually injective or not?

Answer 1

Write $$x^2+ 2xy +y^2 =(x+y)^2, \ \ \ 2x+2y =2(x+y),$$ then your function depends only on $x+y$. So for example, changing $(x,y)$ to $ (x-\epsilon, y+\epsilon)$ do not change the function value. Thus it is not even locally injective.

Answer 2

It is not an injective function. If you assume $f(a,b) = f(c,d)$ for some real numbers $a,b,c,d$ then you only get the equality $a+b=c+d$, which is not guaranteed to be $a = c$ and $b = d$. As mentioned above, taking $(x,y)$ and $(x-\epsilon, y + \epsilon)$ is one of the simplest example.