(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

given a vector fieldv[/B=]Kθ/sθ(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:

[itex]\int[/itex]∇×v.da=[itex]\oint[/itex]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=dss+s dθθ+dzz

The line integral is hence equal to

∫kθdθ with θ runing from 0 to 2[itex]\pi[/itex]

Which has a solution of 2k [itex]\pi[/itex]^{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[itex]\pi[/itex]^{2}meaning the integral will be something like:

∫∫∂(s)kθdθds which evaluates to 2k [itex]\pi[/itex]^{2}as required;

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

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

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