# Is there a name for this 'path' derivative operator?

Let $$\textbf{f}:K\to L^m$$ and $$\textbf{v}:L^m\to M^n$$ such that the derivative of $$\textbf{v}$$ exists for every point $$\textbf{x}=\textbf{f}(t)$$.

The derivative $$D_\textbf{f}\textbf{v}$$ of the function $$\textbf{v}$$ at a point along the path specified by $$\textbf{f}(t)$$ is given by: $$D_\textbf{f}\textbf{v}=\frac{\partial\left(\textbf{v}\circ\textbf{f}(t)\right)}{\partial t}=st\left(\frac{\textbf{v}(\textbf{f}(t+\epsilon))-\textbf{v}(\textbf{f}(t))}{\epsilon}\right)$$ for infinitesimal $$\epsilon$$ ($$st$$ is the standard part)

This should give the change in the vector field $$\textbf{v}$$ experienced by an observer travelling along the path given by $$\textbf{f}$$. (For example, the change in the average current along a particular path across a body of water).

Is there a name for this type of derivative? Does the specification of a particular path make it distinct from the directional derivative?

If $$\mathbf{f}$$ is differentiable, then this is just the directional derivative of $$\mathbf{v}$$ in the direction $$\mathbf{f}'$$. Indeed, that is immediate from the chain rule: the derivative of $$\mathbf{v}\circ\mathbf{f}$$ at a point is the composition of the (total) derivative of $$\mathbf{v}$$ and the derivative of $$\mathbf{f}$$.