Can someone please explain how the following
$$ \textbf{F} \cdot d\textbf{s} = -dV$$
is equivalent to
$$ F_s = -\frac{\partial V}{\partial s}$$
using some intermediate steps. I don't follow this in Goldstein's text. Thanks.
$$\text d f(x)=f'(x)\text d x\equiv\frac{\partial f}{\partial x}\text d x.$$
The second equation you posted is of course only a component representation, the vectors in the first equation are most definately higher dimensional (3 dimensional). And so the derivative in the second equation is chosen such that it goes in the direction of the path, denoted y s. I'm afraid $F_s$ is supposed to denote the "s-component".