The expression for work, dW = F.dx, represents the infinitesimal amount of work done by a force F when it acts over an infinitesimal displacement dx. This formulation arises from the definition of work as the dot product of the force vector and the displacement vector.
The dot product of two vectors, A and B, is given by A·B = |A||B|cos(θ), where |A| and |B| are the magnitudes of the vectors, and θ is the angle between them. In the context of work, the dot product provides a measure of how much of the force is acting in the direction of the displacement.
When the dot product is applied to the work formula, the magnitude of the force is multiplied by the magnitude of the displacement, and then by the cosine of the angle between the two vectors. Since we are considering an infinitesimal displacement, the angle between the force and displacement vectors can be approximated as zero or 180 degrees, depending on whether the force is acting in the same or opposite direction as the displacement.
Now, let's consider the expression dW = x.dF, where x represents displacement and dF represents an infinitesimal force. This expression implies that the displacement is acting on the force, rather than the force acting on the displacement. However, in the context of work, it is the force that is doing the work on an object as it undergoes a displacement. The magnitude of the force determines the amount of work done, while the displacement reflects the distance over which the force is applied.
Therefore, the conventional formulation dW = F.dx is used because it correctly reflects the relationship between the force and the displacement in terms of work. It quantifies the work done by a force when it acts over a given displacement, taking into account the force's magnitude and the distance it acts upon.