In Terry Tao's [notes](http://terrytao.wordpress.com/2008/03/28/285g-lecture-1-ricci-flow/#more-293) on the Poincare Conjecture, he makes a jump I can't understand.



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> From differentiating the identity $g^{\alpha \beta}g\_{\beta \gamma} = \delta^\alpha\_\gamma$ we obtain the variation formula $\frac{d}{dt}g^{\alpha \beta} =-g^{\alpha \gamma}g^{\beta \delta}\dot{g}\_{\gamma \delta}$.
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But I don't understand this leap. Can anyone show me what I'm supposed to be doing? I'm familiar with tensors and the variational calculus but putting the two together is hard for me.



Variation formula of a metric
As mentioned in the comments, once you're comfortable with indices, this is just an exercise in calculus. Differentiating both sides of the identity $g^{\alpha\gamma}g\_{\gamma\zeta} = \delta^{\alpha}\_{\zeta}$ with respect to the parameter $t$, we see that


\begin{align\*}
\frac{d}{dt}\left(g^{\alpha\gamma}g\_{\gamma\zeta}\right) &= \frac{d}{dt}\delta^{\alpha}\_{\zeta}\\
\dot{g}^{\alpha\gamma}g\_{\gamma\zeta} + g^{\alpha\gamma}\dot{g}\_{\gamma\zeta} &= 0\\
\dot{g}^{\alpha\gamma}g\_{\gamma\zeta} &= -g^{\alpha\gamma}\dot{g}\_{\gamma\zeta}\\
\dot{g}^{\alpha\gamma}g\_{\gamma\zeta}g^{\beta\zeta} &= -g^{\alpha\gamma}\dot{g}\_{\gamma\zeta}g^{\beta\zeta}\\
\dot{g}^{\alpha\gamma}\delta\_{\gamma}^{\beta} &= -g^{\alpha\gamma}g^{\beta\zeta}\dot{g}\_{\gamma\zeta}\\
\dot{g}^{\alpha\beta} &= -g^{\alpha\gamma}g^{\beta\zeta}\dot{g}\_{\gamma\zeta}.\\
\end{align\*}


Replacing the index $\zeta$ by the index $\delta$, you get precisely the expression you were looking for. I refrained from using $\delta$ as an index to avoid confusion with the expressions involving $\delta$ in the first and penultimate lines.


