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Stokes theorem in a cylindrical co-ordinates, vector field

  1. Aug 19, 2012 #1
    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:

    [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=ds s +s dθ θ+dz z

    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] 2meaning 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...
     
    Last edited: Aug 19, 2012
  2. jcsd
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