# Stokes theorem in a cylindrical co-ordinates, vector field

1. Aug 19, 2012

### Ratpigeon

1. The problem statement, all variables and given/known data

given a vector field v[/B=]Kθ/s θ (which is a two dimensional vector field in the direction of the angle, θ with a distance s from the origin) find the curl of the field and verify stokes theorem applies to this field, using a circle of radius R around the origin

2. Relevant equations
Stokes Theorem is:

$\int$∇×v.da=$\oint$v . dl
and the curl in cylindrical co-ordinates is:
1/s (∂vz/∂θ-∂vθ/∂z) s+(∂vs/∂z-∂vz/∂s) θ+1/s(∂/∂s (s vθ)-∂vs/∂θ) z
Where vz=0; vs=0; vθ=kθ/s

3. The attempt at a solution
IN cylindrical co-ords; dl=ds s +s dθ θ+dz z

The line integral is hence equal to

∫kθdθ with θ runing from 0 to 2$\pi$

Which has a solution of 2k $\pi$2
However, the curl is zero except for at the centre, where 1/s goes to infinity; so the integral on the other side has a delta function, and the integral will come out at 2k$\pi$ 2meaning the integral will be something like:

∫∫∂(s)kθdθds which evaluates to 2k $\pi$2 as required;

But I'm not sure how to get that integral...

Last edited: Aug 19, 2012
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