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Problem understanding the differential form of the circulation law

  1. Jan 7, 2013 #1
    I've encountered a problem in learning about the curl of a vector field.
    (My learning material is the "Div, Grad, Curl and all that" from H.M. Shey.)

    Introduction to problem:

    The curl of a field F is defined as:
    ∇×F = i (∂Fz/∂y - ∂Fy/∂z) + j(∂Fx/∂z - ∂Fx/∂x) + k(∂Fy/∂x - ∂Fx/∂y)

    He claims the following:
    If we take:
    n.(∇xF) for n = i, j, k
    and they all equal 0 that we can conclude that ∇xF = 0 generally.

    Is it not that we can only conclude that:
    ∂Fz/∂y = ∂Fy/∂z
    ∂Fx/∂z = ∂Fx/∂x and
    ∂Fy/∂x = ∂Fx/∂y

    Or does this conclusion imply the following somehow?

    Thanks .)
  2. jcsd
  3. Jan 7, 2013 #2


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    Homework Helper

    welcome to pf!

    hi havarija! welcome to pf! :smile:

    ∂Fz/∂y = ∂Fy/∂z
    ∂Fx/∂z = ∂Fx/∂x and
    ∂Fy/∂x = ∂Fx/∂y

    is the same as ∇xF = 0 :wink:

    (and, for any vector V, if n.V = 0 for n = i, j, k then V must be 0 !)
  4. Jan 7, 2013 #3
    Re: welcome to pf!

    Now it seems like a ridiculous question :D

    Silly me. But now at least I joined the forum :)
  5. Jan 11, 2013 #4


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    Science Advisor

    Welcome! I love ridiculous questions- I can actually answer some of them!
  6. Jan 14, 2013 #5


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    Gold Member

    if youre reviewing it may be helpful to consider the following definition of curl component:

    [tex]curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr[/tex] where C is a closed loop, A(C) is the area of the loop, vector field F and unit vector u. this is the definition of the u-component of curl.
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