Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Introductory Physics Homework Help
Understanding Christoffel Identity and its Application in Differential Geometry
Reply to thread
Message
[QUOTE="etotheipi, post: 6416174"] Thanks! I'll write out my solution then, using this fact about symmetry in the downstairs indices of the Christoffel symbols,$$\begin{align*} \partial_c g_{ab} &= \Gamma_{bc}^d g_{ad} + \Gamma_{ca}^d g_{bd} \\ \partial_b g_{ca} &= \Gamma_{ab}^d g_{cd} + \Gamma_{bc}^d g_{ad} \\ \partial_a g_{bc} &= \Gamma_{ca}^d g_{bd} + \Gamma_{ab}^d g_{cd} \end{align*}$$It follows that$$\partial_c g_{ab} + \partial_b g_{ca} - \partial_a g_{bc} = 2\Gamma_{bc}^d g_{ad}$$Now I halve both sides, and contract both sides with ##g^{ea}##, making use of the identity ##g^{\alpha \beta}g_{\beta \gamma} = \delta^{\alpha}_{\gamma}##,$$\frac{1}{2} g^{ea} \left(\partial_c g_{ab} + \partial_b g_{ca} - \partial_a g_{bc} \right) = \Gamma_{bc}^d g^{ea} g_{ad} = \Gamma^e_{bc}$$Nice! Muchas gracias señor Ibix! [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Introductory Physics Homework Help
Understanding Christoffel Identity and its Application in Differential Geometry
Back
Top