Line integral around a circle, using polar coordinates

[iAlexN]

Given the force (derived from a potential in planar polar coordinates)

F(p,w) = 2p+sin(w)e_p+cos(w)e_w Where e_p and e_w are unit vectors

How do I calculate the line integral over a circumference that is defined as:
p = 2
0 ≤ w ≤ 2pi

Using the definition of a line integral \int_0^{2pi} \! F(p,w) . \, \mathrm{d}r

What confuses me though is what the "dr" term would be in this case? Do I need to do some form of parametrisation, in that case, how?

Thank you!

 

[FactChecker]

I am confused by your use of "p" and "w" for both the parameters and the unit vector subscripts. In particular, does e_w change when w changes? Or is it fixed?

In any case, since e_p and e_w are unit vectors, you can write them as (cos_p, sin_p) and (cos_w, sin_w). The values of (cos_w and sin_w) may be functions of w.

I think you will have to parameterize the path in order to define the line integral. Since w runs from 0 to 2*pi, I think that is a good parameter to use.