Understanding Partial Derivatives and the Wave Equation

[TheAntithesis]

Homework Statement

Let $f = f(u,v)$ where $u = x+y , v = x-y$
Find $f_{xx}$ and $f_{yy}$ in terms of $f_u, f_v, f_{uu}, f_{vv}, f_{uv}$

Then express the wave equation $\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0$

Homework Equations

Chain rule, product rule

The Attempt at a Solution

I've solved the partial derivatives $f_{xx} = f_{uu) + 2f_{uv} + f_{vv}$ and $f_{yy} = f_{uu) - 2f_{uv} + f_{vv}$

So then $\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0$ is not true unless $f_{uv} = 0$, how am I meant to express it?

 

[Ray Vickson]

Homework Statement

Let $f = f(u,v)$ where $u = x+y , v = x-y$
Find $f_{xx}$ and $f_{yy}$ in terms of $f_u, f_v, f_{uu}, f_{vv}, f_{uv}$

Then express the wave equation $\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0$

Homework Equations

Chain rule, product rule

The Attempt at a Solution

I've solved the partial derivatives $f_{xx} = f_{uu) + 2f_{uv} + f_{vv}$ and $f_{yy} = f_{uu) - 2f_{uv} + f_{vv}$

So then $\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0$ is not true unless $f_{uv} = 0$, how am I meant to express it?

You have just expressed it! f_{uv} = 0.

RGV

 

[TheAntithesis]

I was thinking it couldn't be that simple, apparently it is lol, thanks