# Classical solutions to the wave equation

Suppose $$u(t,x)$$ and $$v(t,x)$$ are $$C^2$$ functions defined on $$\Bbb R^2$$ that satisfy the first-order system of partial differential equations $$u_t$$ = $$v_x$$, $$v_t$$ = $$u_x$$.

Now I have to answer the following questions, but I have no clue how to start.

1: Show that both $$u$$ and $$v$$ are classical solutions to the wave equation $$u_{tt}$$ = $$u_{xx}$$.

2: Which result from multivariable calculus do you need to justify the conclusion?

1. Since the functions belong to the $$C^2$$ class, you have $$u_{tt}=\frac{\partial}{\partial t} u_t= \frac{\partial}{\partial t} v_x = \frac{\partial}{\partial x} v_t=\frac{\partial}{\partial x} u_x=u_{xx}\iff u_{tt} = u_{xx}$$
2. The result in multivariable calculus used to justify the conclusion is the following one: if $$f\in C^2(\Bbb R^n)$$, then the first order partial derivatives to $$f$$ commute each other, i.e. $$\frac{\partial}{\partial x_i}\left( \frac{\partial f(\boldsymbol{x})}{\partial x_j} \right) =\frac{\partial}{\partial x_j}\left(\frac{\partial f(\boldsymbol{x})}{\partial x_i} \right)\quad\forall i,j=1,\ldots,n,\:\:\: i\neq j,\:\: \boldsymbol{x}=(x_1,\ldots,x_n)$$