Integral of a vector with respect to another vector.

[Ludwig]

My physics text gives the following as a general definition of work done by a varying force on a system:
$\sum W = \int (\sum \vec{F}) \cdot d \vec{r}$
Unfortunately, I haven't the faintest idea how to evaluate this. I know how to evaluate an integral with respect to some parameter, but not with respect to another vector. Help?

(Note: I would be particularly interested to know how to evaluate this if given $\vec{F}(t)$ and $\vec{r}(t)$ ).

 

[Orodruin]

The integral is evaluated along a curve $\vec r(t)$ with some curve parameter $t$. It holds that $d\vec r = (d\vec r/dt) dt$ and you can integrate between whatever parameter values you are interested in.

 

[Ludwig]

The integral is evaluated along a curve $\vec r(t)$ with some curve parameter $t$. It holds that $d\vec r = (d\vec r/dt) dt$ and you can integrate between whatever parameter values you are interested in.

So, does that mean I would evaluate this like so? $\int (\sum \vec{F} \cdot \frac{d\vec{r}}{dt})dt$
I.e., evaluate the dot product of the the force and derivative of curve vectors, then integrate with respect to t.

 

[Orodruin]

Yes