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I am having a question on the wikipedia article https://en.wikipedia.org/wiki/Second_covariant_derivative

Using the notation therein I don't get why $(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{b}w^a$.

It is clear that $(\nabla_{u}\nabla_{v}w )^a=(u^c\nabla_{c}v^b\nabla_{b}w)^a$.

What do I miss?

Lucien
  • 1,453
  • The right-hand side of the former equation means the same as the right-hand side of the latter, otherwise you'd be coordinate-dependent. That's the thing I don't like with the index notation for the covariant derivative. It suggests that a coordinate is derived, but actually the coordinate is extracted from the result. – ccorn May 31 '16 at 20:31

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