If we consider a vector field v containing three polar components: $$\boldsymbol{v}=(r, \theta, \phi)$$ giving us the following circular field where r and theta are constant: Sample field (top view)$$\boldsymbol{v}=\boldsymbol{e}_r+\boldsymbol{e}_\theta+\phi\boldsymbol{e}_\phi$$ Is it possible to take an integral whose integrand f is a function of one or more of the components above with respect to our vector field v? For example, lets define f: $$f(r)=r^2$$ and plug f into the following indefinite integral: $$\int f(r)d\boldsymbol{v}$$ If possible, how can this expression be evaluated? I considered expanding v into its corresponding linear combination (2) thinking that might lead somewhere. Maybe taking an integral with respect to a vector could be thought of as being analogous to the directional derivative: $$\nabla_vf(r)$$ If it isn't possible, then could an alternative be to integrate f with respect to the flow curves generated by v? If so, what would that look like?
Any methods would be helpful, thanks in advance!