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Gradient and Curl

  1. Apr 9, 2014 #1
    If the direction of the gradient of f in a point P is the direction of most/minor gradient, so a direction of the curl of f in a point P is the direction of most/minor curl too, correct?

    Also, if the gradient of f in the direction t is given by equation: t, so the curl of f in the direction n is given by equation: ×f·n, correct?
     
  2. jcsd
  3. Apr 10, 2014 #2
    I don't understand what you mean by "minor".

    The direction of the gradient is the direction where [itex]f[/itex] changes the fastest.

    I believe the direction of the curl is the axis about which a sphere would spin, if it were fixed in place and torqued by [itex]\vec{f}[/itex].

    Of course, the "vector" "curl" only works in 3D; it's really just a disguised bivector, which works in any 2+-dimensional space. So in general, the plane of the (bivector) curl would be the plane of rotation for that fixed sphere.

    Interestingly, in 4+ dimensions, there could be multiple such planes simultaneously!

    ---

    As to your second question, I believe you're correct: [itex](\nabla \times \vec{f}) \cdot \vec{n}[/itex] gives the amount of rotation for a fixed axis (fixed along [itex]\vec{n}[/itex]).
     
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