# Calculate curvature by coordinate component method

by PhyPsy
Tags: index notation, riemann tensor
 P: 39 I'm trying to follow the math in Wald's General Relativity where he starts out with the equation for covariant derivative: $\nabla$b$\omega$c = $\partial$b$\omega$c - $\Gamma$dbc$\omega$d He uses that to derive the equation for a double covariant derivative: $\nabla$a$\nabla$b$\omega$c = $\partial$a($\partial$b$\omega$c - $\Gamma$dbc$\omega$d) - $\Gamma$eab($\partial$e$\omega$c - $\Gamma$dec$\omega$d) - $\Gamma$eac($\partial$b$\omega$e - $\Gamma$dbe$\omega$d) Now, using the Riemann tensor definition Rabcd$\omega$d = $\nabla$a$\nabla$b$\omega$c - $\nabla$b$\nabla$a$\omega$c, this equation is derived: Rabcd$\omega$d = $\partial$[a$\partial$b]$\omega$c - $\omega$d$\partial$[a$\Gamma$db]c - $\Gamma$dc[b$\partial$a]$\omega$d - $\Gamma$e[ab]($\partial$e$\omega$c - $\Gamma$dec$\omega$d) - $\Gamma$ec[a$\partial$b]$\omega$e - $\Gamma$ec[a$\Gamma$db]e$\omega$d I know that the term $\partial$[a$\partial$b]$\omega$c cancels, and, due to the symmetry of the metric connection, $\Gamma$e[ab]($\partial$e$\omega$c - $\Gamma$dec$\omega$d) also cancels, but the next step in the book also has a couple other terms cancelled out: $\Gamma$dc[b$\partial$a]$\omega$d, and $\Gamma$ec[a$\partial$b]$\omega$e I don't see how these terms cancel. Can someone help me?

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