Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In Terry Tao's notes on the Poincare Conjecture http://terrytao.wordpress.com/2008/03/28/285g-lecture-1-ricci-flow/#more-293 He makes a jump I can't understand! 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} $ 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.

share|improve this question
1  
Ever heard that if a matrix-valued function $t\mapsto A(t)$ takes non-singular values, then $\frac{d}{dt}A(t)^{-1}=-A^{-1}\frac{dA}{dt}A^{-1}$ ? This is a calculus exercise. –  Denis Serre Mar 1 '12 at 21:22
1  
And this is done most easily using implicit differentiation of $A A^{-1} = I$. –  Deane Yang Mar 1 '12 at 21:40
1  
It's also worth noting that the two comments above are equivalent to just differentiating $g^{\alpha\beta}g_{\beta\gamma} = \delta^\alpha_\gamma$ with respect to the variation parameter and then solving for $\dot{g}^{\alpha\beta}$ by multiplying the differentiated equation by $g^{\eta\alpha}$. –  Deane Yang Mar 2 '12 at 17:35
    
Ok so it was a lot simpler than I thought. –  Peadar Coyle Mar 3 '12 at 11:03

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.