I was reading up on covariant derivatives and came across this document. On the second page it says:
We define a procedure called parallel transport by defining a vector $\vec A (\lambda)$ along each point of the curve in such a way that $DA^\mu/d\lambda=0$: $$\nabla_V \vec A =0 \iff \text{parallel transport of } \vec A \text{ along } \vec V $$
My question is what is the difference (if any) between $D A^\mu/d\lambda$ and $\nabla_V \vec A$, if they are the same please can you explain why the use of different notations.
The relation between these two expressions is illustrated in eq. (22) of the document linked above:
$$\frac{d \vec A}{d \lambda} \equiv \frac{D A^{\mu}}{d \lambda} \vec e_{\mu} \equiv \nabla_V \vec A \tag{22}$$
where Einstein summation over index $\mu$ is implied, and $\nabla_V$ denotes the directional derivative along the given curve $\mathbf x[~\lambda~]$ with tangent vector $\vec V$.
Accordingly, $\frac{D A^{\mu}}{d \lambda}$ denotes the coefficient of (any) one direction component of the full derivative vector:
$$\frac{D A^{\mu}}{d \lambda} \equiv \frac{(\nabla_V \vec A) \cdot \vec e_{\mu}}{\vec e_{\mu} \cdot \vec e_{\mu}} = (\nabla_V \vec A) \cdot \vec e_{\mu}$$
for any unit base vector $\vec e_{\mu}$.