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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?

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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 $\mathbf{f}$.

This fact is in fact absolutely central to the theory of differentiable manifolds. On an abstract manifold, we don't have privileged "straight" paths (paths which are straight in one coordinate chart will be curved in another), so all directional derivatives of functions on manifolds are defined as derivatives along curves in this way.

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