differentiation - $delta$ differential notation

Various textbooks that I am currently consulting (including Spacecraft Dynamics and Control An Introduction - Anton H.J. De Ruiter | Christopher J. Damaren | James R. Forbes Section 1.4, page 32) use $\delta$, not $d$ or $\partial$ to express an infinitesimal quantity. In the context of the reference text above, the symbol is used as part of a general definition for the derivative of a vector. Specifically, given

$$\mathbf{\vec{r}}=\mathcal{\vec{F^{T}_{1}}}\mathbf{\vec{r_{1}}}$$

where

$$\mathcal{\vec{F^{T}_{1}}}$$

is a vectrix

$$[\mathbf{\vec{x_1}} \phantom{s} \mathbf{\vec{y_1}} \phantom{s} \mathbf{\vec{z_1}}].$$

The time derivative of the vector is defined as

$$\dot{\mathbf{\vec{r}}}\triangleq \lim_{\delta t\to 0}\frac{\delta\mathbf{\vec{r}}}{\delta t}.$$

In this context, what is the difference between $\frac{\delta\mathbf{\vec{r}}}{\delta t}$ and $\frac{d \mathbf{\vec{r}}}{d t}$ ?

Note about thermodynamic use of $\delta$: My understanding is that $\delta$ is used in thermodynamic equations to express path dependence of a scalar quantity such as heat or work. What does it mean in the context of an abstract physical vector?

Answer

1. 
   

   Typically, $\frac{d}{d(\ldots)}$ is a (total) derivative, $\frac{\partial}{\partial(\ldots)}$ is a partial derivative, and $\frac{\delta}{\delta(\ldots)}$ is a functional/variational derivative. See also e.g. this & this Phys.SE posts and links therein.

   
   


2. 
   

   In Ref. 1 the symbol $\frac{d\mathbf{\vec{r}}}{d t}$ denotes the derivative, while $\frac{\delta\mathbf{\vec{r}}}{\delta t}$ denotes the difference quotient. A more common notation for the latter is $\frac{\Delta\mathbf{\vec{r}}}{\Delta t}$.

   
   

References:

1. A.H.J. De Ruiter, C.J. Damaren & J.R. Forbes, Spacecraft Dynamics and Control: An Introduction; Section 1.4, p. 32.