Why does $W = \int F\cdot \mathrm{d}s$ rather than $\int s\cdot\mathrm{d}F$

Question

Conventionally, infinitesimal work is defined as $\delta w = F\cdot ds$ and its integral as the work $$w(P_1 \to P_2) = \int_{P_1}^{P_2} F\cdot ds \tag{1}.$$

The word work, of course, can be assigned to anything, it is just a definition, but does the other infinitesimal quantity defined as $\delta w^* = s\cdot dF$ or its corresponding integral between the same points $P_1,P_2$: $$w^*(P_1 \to P_2) = \int_{P_1}^{P_2} s\cdot dF \tag{2}$$ have physical significance (meaning), and if yes what?

Answer 1

I will assume that, following the usual definition of work, $F$ is a force and $s$ is displacement from a fixed origin.

The problem with assigning a physical meaning to $s \cdot dF$ is that the value of $s$ depends on where you place the origin of your co-ordinate system (whereas $ds$, being a difference of two $s$ values, does not). A physically meaningful quantity should not be co-ordinate dependent.