What does $\delta/\delta t$-derivative represent in tensor calculus?

Question

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the aforesaid text defines the acceleration $\mathbf{A}$ of a particle on a surface as

$$\mathbf{A} = \frac{\delta V^{\alpha}}{\delta t}\mathbf{S}_{\alpha} + \mathbf{N}B_{\alpha\beta}V^{\alpha}V^{\beta},$$

where

$$\frac{\delta V^{\alpha}}{\delta t} = \frac{d V^{\alpha}}{d t}+\Gamma^{\alpha}_{\beta\gamma}V^{\beta}V^{\gamma}.$$

I already know that the RHS of the second equation is the contravariant component of the acceleration vector $\mathbf{A}$. But, I don't get why $\frac{\delta V^{\alpha}}{\delta t}$, and not simply $\frac{d V^{\alpha}}{d t}$, is multiplied to $\mathbf{S}_{\alpha}$ in the first equation. In particular, given $\mathbf{V} = V^{\alpha}\mathbf{S}_{\alpha}$, one may write

$$\begin{align} \mathbf{A} &= \frac{d}{d t}[V^{\alpha}\mathbf{S}_{\alpha}]\\&= \mathbf{S}_{\alpha}\frac{d V^{\alpha}}{d t}+V^{\alpha}\frac{d \mathbf{S}_{\alpha}}{d t}\\&= \mathbf{S}_{\alpha}\frac{d V^{\alpha}}{d t} + V^{\alpha}\frac{d \mathbf{S}_{\alpha}}{d S^{\beta}}\frac{d S^{\beta}}{d t}\\&=\mathbf{S}_{\alpha}\frac{d V^{\alpha}}{d t} + \mathbf{N}B_{\alpha\beta}V^{\alpha}V^{\beta}. \end{align}$$

So,

1- can one explain what is wrong in my derivation above?

2- generally, why do we define $\delta/\delta t$-derivative?

Answer 1

If $t$ is parameter which varies along some time-like or null geodesic, and say the tangent vector at each point on the geodesic is given by $V^a=\frac{dx^a}{dt}$, then the acceleration vector is simply defined as the quantity $$A^a=V^b\nabla_bV^a=V^b\partial_bV^a+\Gamma^a_{bc}V^bV^c$$ Check the first term : $$V^b\partial_bV^a=\frac{\partial x^b}{\partial t}\frac{\partial V^a}{\partial x^b}=\frac{\partial V^a}{\partial t}$$ Now, you can readily identify $\frac{\delta}{\delta t}$ with $V^a\nabla_a$ operator which measures the rate of change along a given geodesic. You can check the 3+1 decomposition technique where this definition is commonly used, see for instance https://arxiv.org/abs/gr-qc/0503113.