Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$. Let $r ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) , s ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y )$ and $t ( x , y ) = \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y )$ so that, for all $( x , y ) \in \mathbb { R } ^ { 2 }$, $r ( x , y ) > 0$ and $r ( x , y ) t ( x , y ) - s ( x , y ) ^ { 2 } = 1$.
[IV.B.1)] Show that there exist two functions $\varphi$ and $\psi$ in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ such that $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \left\{ \begin{array} { l }
\varphi ( u ( x , y ) , v ( x , y ) ) = x - \frac { \partial f } { \partial x } ( x , y ) \\
\psi ( u ( x , y ) , v ( x , y ) ) = - y + \frac { \partial f } { \partial y } ( x , y )
\end{array} \right.$$
[IV.B.2)] Compute $\frac { \partial \varphi } { \partial u } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \varphi } { \partial v } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \psi } { \partial u } ( u ( x , y ) , v ( x , y ) )$ and $\frac { \partial \psi } { \partial v } ( u ( x , y ) , v ( x , y ) )$ (which we will abbreviate as $\frac { \partial \varphi } { \partial u } , \frac { \partial \varphi } { \partial v } , \frac { \partial \psi } { \partial u }$ and $\frac { \partial \psi } { \partial v }$) in terms of $r ( x , y ) , s ( x , y )$ and $t ( x , y )$ (which we will abbreviate as $r , s$ and $t$).
[IV.B.3)] Show that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are bounded on $\mathbb { R } ^ { 2 }$.
[IV.B.4)] Show, using the first part, that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are constant.
[IV.B.5)] Deduce that $r , s$ and $t$ are constant.
Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$:
$$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$
For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$. Let $r ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) , s ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y )$ and $t ( x , y ) = \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y )$ so that, for all $( x , y ) \in \mathbb { R } ^ { 2 }$, $r ( x , y ) > 0$ and $r ( x , y ) t ( x , y ) - s ( x , y ) ^ { 2 } = 1$.
\begin{enumerate}
\item[IV.B.1)] Show that there exist two functions $\varphi$ and $\psi$ in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ such that
$$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \left\{ \begin{array} { l }
\varphi ( u ( x , y ) , v ( x , y ) ) = x - \frac { \partial f } { \partial x } ( x , y ) \\
\psi ( u ( x , y ) , v ( x , y ) ) = - y + \frac { \partial f } { \partial y } ( x , y )
\end{array} \right.$$
\item[IV.B.2)] Compute $\frac { \partial \varphi } { \partial u } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \varphi } { \partial v } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \psi } { \partial u } ( u ( x , y ) , v ( x , y ) )$ and $\frac { \partial \psi } { \partial v } ( u ( x , y ) , v ( x , y ) )$ (which we will abbreviate as $\frac { \partial \varphi } { \partial u } , \frac { \partial \varphi } { \partial v } , \frac { \partial \psi } { \partial u }$ and $\frac { \partial \psi } { \partial v }$) in terms of $r ( x , y ) , s ( x , y )$ and $t ( x , y )$ (which we will abbreviate as $r , s$ and $t$).
\item[IV.B.3)] Show that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are bounded on $\mathbb { R } ^ { 2 }$.
\item[IV.B.4)] Show, using the first part, that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are constant.
\item[IV.B.5)] Deduce that $r , s$ and $t$ are constant.
\end{enumerate}