Suppose we have a smooth function $u(t,x)$ such that $u_{tt}=u_{xx}$
I want to construct a function $v(t,x)$ such that $v_{t}=u_{x}$ and $v_{x}=u_{t}$
My approach:
Let $v(t,x)$ be a function.
Let $v_{tx}=u_{xx}$: this implies $v_{t}=u_{x}$
Let $v_{xt}=u_{tt}$: this implies $v_{x}=u_{t}$
Since $u_{tt}=u_{xx}$, then we have $v_{tx}=v_{xt}$
$\therefore$ Our function $v(t,x)$ must pertain to $C^2$, its second partial derivatives must be continuous.
Is this correct?