# Integral of a vector with respect to another vector.

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
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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.

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
Staff Emeritus