Originally Posted by

**mukmar** I need to prove:

$\displaystyle \displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x - t) \frac{\partial^2\phi}{\partial x^2}(x, t) dx dt = \displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x - t) \frac{\partial^2\phi}{\partial t^2}(x, t) dx dt$

Where $\displaystyle f(x - t)$, is a function of $\displaystyle x$ and $\displaystyle t$, and is merely piecewise continuous, so it doesn't have to be differentiable at certain points, and $\displaystyle \phi(x,t)$ is an arbitrary function also of $\displaystyle x$ and $\displaystyle t$, which is infinitely differentiable and has compact support.

My Attempt:

I tried to change variables by using $\displaystyle u = x - t$ and $\displaystyle v = x + t$, and I later realized that it basically became an exercise in trying to prove $\displaystyle \frac{\partial^2\phi}{\partial x^2} = \frac{\partial^2\phi}{\partial t^2}$ which didn't work out.

Can anyone give me a hint of some kind on how to try and proceed? Any suggestions would be greatly appreciated. Thanks in advance.