Variation of a tensor expression with indices

[spaghetti3451]

Say I want to find $\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)$.

Is the following alright: $\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)$?

 

[Orodruin]

Say I want to find $\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)$.

Is the following alright: $\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)$?

No. You have broken out terms of a sum and it is no longer clear what is being summed with what. I suggest you use the chain rule $\delta f(x) = f'(x) \delta x$.

 

[spaghetti3451]

Thanks! I've got it.