General Work Functions: Validity Check

[Ludwig]

I'm trying to derive a general work function (provided force and displacement vector-valued functions). Below are my best guesses. Can someone let me know whether these are valid?

Rigid-System:
$\sum W = \int \left ( \sum \vec{F}(t)\cdot \vec{r}\,'(t) \right ) dt$

Deformable-system (n-forces):
$\sum W = \sum_{k=0}^{n} \left (\int ( \vec{F}_{k}(t)\cdot \vec{r}\,'(t)\,) dt \right)$

 

[Simon Bridge]

$F(t)\cdot\vec r(t) = W(t)$

So $\sum W = \sum \vec F(t) \cdot \vec r(t)$ notice: no integral on the RHS.

Try starting from: $\text{d}W = \vec F_{tot} \cdot \text{d}\vec r$ and change variable to time.