Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Curl and its relation to line integrals

  1. Feb 20, 2013 #1

    joshmccraney

    User Avatar
    Gold Member

    hey all

    i know and understand the component of curl/line integral relation as: [tex]curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr[/tex] where we have vector field [itex]F[/itex], [itex]A(C)[/itex] is the area of a closed boundary, [itex]u[/itex] is an arbitrary unit vector, [itex]dr[/itex] is an infinitely small piece of curve [itex]C[/itex]

    my question is, how does this definition change if i have, say [itex]curlF\cdot {x}[/itex] versus [itex]curlF\cdot {z}[/itex] where [itex]x[/itex] and [itex]z[/itex] are the unit vectors in the standard cartesian system.

    thanks for the feedback! you guys/girls are amazing!
     
  2. jcsd
  3. Feb 20, 2013 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I don't understand your question. First what "definition" are you talking about? The formula you give is not a definition. Second, you are given a formula for [itex]curl F\cdot u[/itex] where u can be any unit vector- but there is no reference to c on the right side- they cannot be equal. Did you mean that u is the unit vector perpendicular to the plane of C? But you did not require that C be a planar curve.
     
  4. Feb 22, 2013 #3

    joshmccraney

    User Avatar
    Gold Member

    I read information about this on a vector analysis course page by the university of minnesota, where they said the line integral above was the way one formally defines curl. is this incorrect?

    yes, apologies here. [itex]C[/itex] is a planar closed curve around some point in space orthogonal to [itex]u[/itex]
     
  5. Feb 23, 2013 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    It's not really incorrect, since it's equivalent to the usual definition (if everything is smooth enough). But it's not the standard definition. Usually textbooks defines curl totally differently. The standard definition is: http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx

    I actually do like your limit definition better since it is way more intuitive.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook