Polar Co-Ordinates/Closed Curves
Hi all, please bear with me, I am new here. I have a problem with polar co-ordinates and closed curves. I am unsure how to start the following:
Given u(r(o))=Rsin(o)R+Rsin^2(o)cos^2(o)o. Calculate the integral around the closed curve C of u.dr. (Here, o=theta)
I thought that maybe, since the curve had been written as u(r(o)), then there was some parametrisation here. However, u is in polar co-ordinates, so I couldnt figure out what this would be. ie: I would have an integral of the form:
But I cant see how this helps, since there is still a product of sin squared cos squared in the second bit, even when you dot it with dr/do, which isnt gonna be easy to integrate!
Further, I am asked to determine the integral around a surface S of curl u dot normal.dS, with the surface S given by: r: x^2+y^2=1, z=0, with normal n=z.
For this part, I know that if I use Stokes Theorem, I get an integral around a closed curve of u.dr, ie. the same as in the first part. However, will the value of this integral be 2*Pi (basically, the diameter of the circle is 1, then just find the circumference through Pi*d)?
Any help on these would be great.