What happens when this turns to $dx$?

Question

I have this equation: $$ ds^2=c^2dt^2-dx^2-dy^2-dz^2. $$ And I've also been given $$ x=x'\cos(\Omega t)-y'\sin(\Omega t), $$ which I need to substitute into the first equation. I've squared $x$ to get $$ x^2=x'^2\cos^2(\Omega t)+y'^2\sin^2(\Omega t)-2x'y'\sin(\Omega t)\cos(\Omega t), $$ but I can't think of what happens when this turns into a $dx^2$ so I can make the substitution. Thanks for any help; I'm very stuck.

Answer 1

Use the chain rule

$$ {\rm d}x = \frac{\partial x}{\partial x'}{\rm d}x' + \frac{\partial x}{\partial y'}{\rm d}y' + \frac{\partial x}{\partial t}{\rm d}t $$

$$ {\rm d}x = ({\rm d}x' - \Omega y' {\rm d}t) \cos( \Omega t) - ( {\rm d}y'+ \Omega x' {\rm d}t )\sin (\Omega t) $$

and then ${\rm d}x^2 = ({\rm d}x)^2 = {\rm d}x\, {\rm d}x = \ldots $