- #1

- 952

- 113

- Summary:
- Definition of covariant derivative as a tensor through the limiting process of a fraction

Hi,

I've been watching lectures from XylyXylyX on YouTube. I believe they are really great !

One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a collection of vector components (living in the tangent space at point Q) and the denominator is a bunch of real numbers.

My point is: to be a (1,1) tensor it has to transform accordingly. The numerator is a vector and thus its components transform as such; what about the denominators ##\delta x^{\alpha}## ? I believe that the inverse of them have really to be the components of a co-vector

Is that the case ?

I've been watching lectures from XylyXylyX on YouTube. I believe they are really great !

One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a collection of vector components (living in the tangent space at point Q) and the denominator is a bunch of real numbers.

My point is: to be a (1,1) tensor it has to transform accordingly. The numerator is a vector and thus its components transform as such; what about the denominators ##\delta x^{\alpha}## ? I believe that the inverse of them have really to be the components of a co-vector

Is that the case ?

Last edited: