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(a) State the Implicit Function Theorem in the most general way that you know

(b) Let $\Sigma$ the set of $2 \times 2$ matrices with determinant zero. Show that if $0 \neq M \in \Sigma$, then there is a neighborhood $V \ni M$ such that $\Sigma \cap V$ is a parameterized surfaces.

(c) Write explicitly a parameterization for $\Sigma \cap V$ with $M = \left(\begin{array}{cc}1 & 1 \\ 1 & 1\end{array} \right)$ and $V \ni M$ of your choice.

Attempt.

(a) Let $F: \mathbb{R}^{n} \times \mathbb{R}^{m} \to \mathbb{R}^{m}$ be a $C^{1}$ function. Suppose that $F(a,b) = 0$ and $\displaystyle \det\left(\frac{\partial}{\partial y}F(a,b)\right) \neq 0$. So there is $a \in A$ and $b \in B$ open sets and a $C^{1}$ function $f: A \to B$ such that

  • $b = f(a)$,
  • $F(x,f(x)) = 0$ for each $x \in A$

and $f$ is a

  • $\frac{\partial}{\partial x_{i}}f(x,y) = -\left(\frac{\partial}{\partial y}F(a,b)\right)^{-1}\left(\frac{\partial}{\partial x_{i}}F(a,b)\right)$.

(b) Consider the function $F: (\mathbb{R}^{3}\times \mathbb{R}) \to \mathbb{R}$ given by $F(X) = \det(X) = a_{11}a_{22} - a_{12}a_{21}$ for $$X = \left(\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right).$$

We have $F \in C^{1}$, $F(M) = F(m_{11},m_{12},m_{21},m_{22}) = 0$ and $\det\left(\frac{\partial}{\partial a_{22}}F(M)\right)^{-1} \neq 0$. By the Implicit Function Theorem, $F^{-1}(0)\cap(V \times U) = (\Sigma\cap V)\times(\Sigma\cap U)$ is the graph of a continuously differentiable function $f: M \in V \to F(M) \in U$. But the graph of a $C^{1}$ function is a surface. Therefore, $\Sigma \cap V$ is a parameterized surface.


(c) I can write $f$ as $\displaystyle a_{22} \mapsto \frac{a_{12}a_{21}}{a_{11}}$. But I dont know how to choice $V$ and find a parameterization.

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(a) I believe you are missing some subtleties. I'd recommend Krantz' book

(b) We have $\nabla F(X) = (a_{22},a_{21},a_{12},a_{11})$. This ensures that $\nabla F(M)\neq 0$ for any $M\neq 0$, Hence you may apply the Implicit Function Theorem. In your attempt you covered just the case $M=(m_{ij})$ with $m_{11}\neq 0$. However, the other cases follow analogously.

(c) you have that $\displaystyle \frac{\partial F}{\partial a_{11}}(M) = m_{22}=1 \neq 0$. Then you can choose the parameterization: $$ (a_{12}, a_{21}, a_{22}) \mapsto \begin{pmatrix} \frac{a_{12}a_{21}}{a_{22}} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} $$ And you can take $V= \{X \mid a_{22} \neq 0\}$. Indeed, $V = \{(x,y,z) \in \mathbb{R}^3 \mid z\neq 0\}\times \mathbb{R}$ and $\Sigma \cap V$ is the graph of $\displaystyle f\colon \{(x,y,z) \in \mathbb{R}^3 \mid z\neq 0\} \to \mathbb{R}$ given by $\displaystyle f(x,y,z) = \frac{xy}{z}$.

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  • $\begingroup$ Thank you, its very helpful! I will check the reference. $\endgroup$ – Lucas Corrêa Feb 21 at 3:54

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