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Antic_Hay

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## Homework Statement

Show that the electromagnetic wave equation

[itex]\frac{\partial^{2}\phi}{\partial x^{2}} +

\frac{\partial^{2}\phi}{\partial y^{2}} +

\frac{\partial^{2}\phi}{\partial z^{2}} -

\frac{1}{c^2}\frac{\partial^{2} \phi}{\partial t^2} [/itex]

is invariant under a Lorentz transformation.

## Homework Equations

Lorentz Transformations:

[itex]x' = \frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}}[/itex]

[itex]t' = \frac{t - \frac{v}{c^2}x}{\sqrt{1 - \frac{v^2}{c^2}}}[/itex]

[itex]y' = y[/itex]

[itex]z' = z[/itex]

## The Attempt at a Solution

Well I know exactly what I'm supposed to do here, transform the equations from x to x', y to y' etc. Then rearrange terms and show that the wave equation with x,y,z, and t is equal to the wave equation with x', y', z', and t'.

I understand the method is to get [itex]\frac{\partial x'}{\partial x}[/itex], then use the chain rule ( [itex]\frac{\partial\phi}{\partial x} = \frac{\partial\phi}{\partial x'}\frac{\partial x'}{\partial x}[/itex] ), and similarly for t -> t', then a bit of simple algebra and the answer should pop out the other end.

My only problem is that I have no idea what [itex]\frac{\partial x'}{\partial x}[/itex] is...in fact, I do know, since I have been told, that [itex]\frac{\partial x'}{\partial x} = \frac{1}{\sqrt{1 - v^2/c^2}}[/itex], but I have no idea how that follows from the Lorentz transforms, could someone give me a hint in deriving it?

(My attempt was simply saying [itex]\partial x' = \frac{\partial x - v\partial t'}{1 - v^2/c^2}[/itex], but then dividing that by [itex]\partial x'[/itex] I'd get [itex]\frac{1 - \frac{\partial t'}{\partial t}}{\sqrt{1-v^2/c^2}}[/itex], which isn't right...

As you can tell, I'm not really very comfortable with partial derivatives or differential calculus, so I'm probably doing something really silly in that derivation above...

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