I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators [itex]\partial_{a}[/itex] are dependent on the coordinate system one chooses and thus not naturally associated with the structure of the manifold, therefore we introduce a new derivative operator [itex]\nabla_{a}[/itex] which fulfils this criteria. Intuitively, the covariant derivative is constructed in such a way as to remove any artefacts arising from any particular coordinate system, such that the rate of change of a tensor field in a direction along the manifold is itself a tensor field. Is this correct?(adsbygoogle = window.adsbygoogle || []).push({});

He then goes on to say that a vector [itex]\mathbf{v}^{a}[/itex] given at each point along a curve [itex]C[/itex] (with tangent vector [itex]\mathbf{t}^{a}[/itex]) is said to beparallely transportedas one moves along the curve if the equation $$\mathbf{t}^{a}\nabla_{a}\mathbf{v}^{b}=0$$ is satisfied along the curve.

Intuitively, is this a statement that a vector [itex]\mathbf{v}^{a}[/itex] is parallely transported along the curve if it remainsconstantas it moves along the curve, this can be translated into the mathematical statement that its directional derivative along the curve vanishes, i.e. [itex]\mathbf{t}^{a}\nabla_{a}\mathbf{v}^{b}=0[/itex]. Is this correct?

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# Parallel transport and the covariant derivative

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