What is the difference between derivative and covariant derivative?
I realized that $\dfrac {d^2x^a}{ds^2}\ne 0$ while $\dfrac {D^2x^a}{Ds^2}=0$.
Why is it like this?
What is the difference between derivative and covariant derivative?
I realized that $\dfrac {d^2x^a}{ds^2}\ne 0$ while $\dfrac {D^2x^a}{Ds^2}=0$.
Why is it like this?
In this case, it may be a bit easier to explain in terms of a manifold in an embedding.
If $\underline P$ is the projection operator onto the manifold, $\nabla$ is the vector derivative of the embedding and $\partial = \underline P(\nabla)$ is the projection of the vector derivative into the manifold, then the covariant derivative $D$ obeys
$a \cdot D A = \underline P(a \cdot \partial A)$
So that it lies entirely in the manifold, guaranteed.
In particular, the covariant derivative and projected derivative are related by
$a \cdot D A = a \cdot \partial A + \underline S(a) \times A$
where $\underline S(a)$ is the shape tensor. This is why an expression of partial derivatives can be nonzero but the covariant derivative can be zero--the shape tensor term comes into play.