What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$?

Question

Let $f:\mathbb{R}^3 \to \mathbb{R}^3$ be $$f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right).$$

My question is what restrictions are necessary on $i,h,j,k,m,n$ for there to be a neighborhood around a point $(x,y,z)$ that maps 1:1 onto an open set of the range space? Is the global mapping ever 1:1 and onto?

In understanding this function and how to apply the inverse function theorem, I'm not sure how to deal with the coefficients here. Is there a good textbook/resource that covers examples similar to this?

Answer 1

Given: $$\begin{gathered} f:{\mathbb{R}^3} \to {\mathbb{R}^3} \hfill \\ f({\text{x}},{\text{y}},{\text{z}}) = \frac{1}{2}\left( {\begin{array}{*{20}{c}} {\alpha \cdot {x^2} + \beta \cdot {y^2}} \\ {\gamma \cdot {y^2} + \delta \cdot {z^2}} \\ {\varepsilon \cdot {x^2} + \varphi \cdot {z^2}} \end{array}} \right) \hfill \\ \end{gathered} $$ Then: $$Df(x) = \left( {\begin{array}{*{20}{c}} {\alpha \cdot x}&{\beta \cdot y}&0 \\ 0&{\gamma \cdot y}&{\delta \cdot z} \\ {\varepsilon \cdot x}&0&{\varphi \cdot z} \end{array}} \right)$$ and $$\det (Df(x)) = (\alpha \gamma \varphi + \beta \delta \varepsilon ) \cdot xyz$$ So $\det (Df(x)) \ne 0$ if $\alpha \gamma \varphi + \beta \delta \varepsilon \ne 0$ and $xyz \ne 0$. In this case everything is regular.

Answer 2

I don't know about resources on this, but think about the conditions for the inverse function theorem to hold. What's the one equationy condition you need?

That condition is that the point you consider is not a critical point. In other words, the Jacobian determinant is non-zero at that point. See where that takes you. Hope that helps!