1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Curl is a measure of the tendency of a vector field

  1. Feb 22, 2012 #1
    [itex]\nabla[/itex][itex]\times[/itex]grad(f) is always the zero vector. Can anyone in terms of physical concepts make it intuitive for me, why that is so. I get that the curl is a measure of the tendency of a vector field to rotate or something like that, but couldn't really assemble an understanding just from that.
     
  2. jcsd
  3. Feb 22, 2012 #2

    lanedance

    User Avatar
    Homework Helper

    Re: curl(grad(f))

    do you know stokes theorem, which for any vector field g is something like
    [tex]
    \int_A \nabla \times \textbf{g} \cdot \textbf{dA}
    = \oint_{\partial A} \textbf{g} \cdot \textbf{dr}
    [/tex]

    it converts an integral over a surface into an inetgral over the boundary. no conside [itex] g = \nabla f [/itex], any closed path integral will be zero - why?
     
  4. Feb 22, 2012 #3

    lanedance

    User Avatar
    Homework Helper

    Re: curl(grad(f))

    also worth understanding conservative vector fields here
     
  5. Feb 22, 2012 #4
    Re: curl(grad(f))

    Physically, I'm not sure, but isn't mathematically enough ?
    grad(f) is [itex]\nabla[/itex]f so you take the vector product of a vector ([itex]\nabla[/itex]) with one // to itself ([itex]\nabla[/itex]f) so it will be 0 by definition of the vector product. (sorry if it does not help)

    cheers..
     
  6. Feb 22, 2012 #5
    Re: curl(grad(f))

    i'm not especially sure on this, but looking at a drawing, i see that a vector field has rotation near a point x if that point is surrounded by closed loops. but then that means there isn't any directed gradient. conversely, if there is a direction for the gradient field, then there aren't any closed loops to measure rotation.
     
  7. Feb 22, 2012 #6
    Re: curl(grad(f))

    I know Stokes theorem, but that's just a mathematical theorem, which I by the end of the day then would want to understand intuitively. I do understand that a conservative field is a vector field, which is the gradient of a scalar field. It is then easily shown that the path integral between 2 points is independent of the road taken.
    I just wanted the physical interpretation, and I haven't got it from those answers I'm afraid.
    In other words: Further help needed! :)
     
  8. Feb 22, 2012 #7
    Re: curl(grad(f))

    The gradient 'sort of tells you how f is changing in a certain direction'
    the curl, 'sort of tells you how much it changes in an orthogonal direction'
    once you have limited the changes in the one direction of the grad, the variation that would go in the orthogonal direction of what is left (the curl of the grad) is, well, zero :)
     
  9. Feb 22, 2012 #8

    lanedance

    User Avatar
    Homework Helper

    Re: curl(grad(f))

    well stokes theorem tells you the curl is similar to the path integral around a loop as you shrink the loop to zero length (orentated in a certain direction)

    the gradient of a function is conservative, so the integral around any closed loop will be zero, and isf c is any point onteh closed loop, you can represent as follows
    [tex]
    \int_A \nabla \times \nabla f \cdot \textbf{dA}
    = \oint_{\partial A} \nabla f \cdot \textbf{dr} = f(c)-f(c)=0
    [/tex]
     
  10. Feb 22, 2012 #9

    kai_sikorski

    User Avatar
    Gold Member

    Re: curl(grad(f))

    To me this picture helps a lot to understand Stokes theorem

    250px-Stokes.png

    It just shows how if you wanna find the circulation (RHS of stokes theorem) for the larger region, that's the same as adding up the contributions from the smaller rectangles because all the contributions from sides not on the boundary of the region cancel.

    So to understand Stokes theorem you just need to understand why it would work for an infinitesimal rectangle. There I'm afraid you just need to write out the terms using a Taylor expansion and see that it works. Some things don't necessarily have a nice intuitive explanation.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Curl is a measure of the tendency of a vector field
  1. Curl of a vector field (Replies: 11)

  2. Curl of a vector field (Replies: 6)

Loading...