How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ?
To give context, this comes from: Dirac's Theory of General Relativity p19: https://i.stack.imgur.com/2fall.jpg
I'm not comfortable with proofs regarding variations of functions. They always look intuitively obvious. This looks like the chain rule. But how would In prove it rigorously? I see varitaions come up a lot in mathematical physics, but ive never covered them in detail in mathematics.
When I look at my old notes I wrote: $d(x^2)= 2x dx$. Then substituted $ds$ for $x$ and took $dx= \delta (ds)$ That seems a bit sketchy to me now.
This is from page 19. General Theory of Relativity by Dirac.