Why is small work done always taken as d W = F ⋅ d x and not d W = x ⋅ d F ?

Question

Why is small work done always taken as    d  W  =  F  ⋅  d  x and not   d  W  =  x  ⋅  d  F ?

Steven Bates · Accepted Answer

You can see this from the second Newton&#039;s law:  m                    x            ¨        =      F    (      x    )Now I would like to integrate this equation of motion with respect to time, to arrive at the energy conservation. To do so I multiply both sides with                     x            ˙      :  m                    x            ¨        ⋅                    x            ˙        =      F    (      x    )  ⋅                    x            ˙      and finally integrate:  m  ∫  d  t                    x            ¨        ⋅                    x            ˙        =  ∫  d  t      F    (      x    )  ⋅                    x            ˙      The l.h.s. gives me the kinetic energy. The r.h.s. gives me exactly the integral in question:      1    2    m                              x                ˙              2    =  ∫  d      x    ⋅      F    (      x    )So the work done by the force is the kinetic energy of the particle (up to an integration constant representing its total energy).

Why is small work done always taken as dW=Fdx and not dW=xdF?

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