Hello! I have a question regarding the tangentvector belonging to a parameterized curve, at a certain point p.(adsbygoogle = window.adsbygoogle || []).push({});

Lee has argued in his book 'Introduction to Smooth Manifolds' that the gradient earlier encountered as:

[itex](\partial_\mu)f ,[/itex]

is not a tangentvector since it is not always coordinate independent in the tangentvector basis when we cannot define a metric. Instead these components är coordinateindependent in the dual vector basis [itex]dx^\mu[/itex].

When trying to describe the velocity of a particle moving along a parameterized curve these components are the ones to use, so...

Does this mean that for manifolds without any metric, the velocity of, lets say a particle, always should be described as a covector (the differential df) or that the velocity in manifolds without any defineable metric is coordinate independent.. :/ Am I talking about two different things here maybe?

A second question is regarding the notation of vectors i a basis by:

[itex]\vec{x} = x^\mu \hat{e}_\mu,[/itex]

maybe this is a stupid question, but I just want to be sure. Our components x^\mu is not a contravariant vector here, right? The index is instead just put in the upper position for the Einstein summation convention?

Thanks for your support in my studies! :)

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# Vectors and covectors

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