Understanding Partial Derivatives and the Wave Equation

[TheAntithesis]

Homework Statement

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

Then express the wave equation [itex]\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0[/itex]

Homework Equations

Chain rule, product rule

The Attempt at a Solution

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

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

 

[Ray Vickson]

Homework Statement

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

Then express the wave equation [itex]\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0[/itex]

Homework Equations

Chain rule, product rule

The Attempt at a Solution

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

So then [itex]\frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0[/itex] is not true unless [itex]f_{uv} = 0[/itex], 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