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Definition of curl

  1. Oct 20, 2015 #1

    Titan97

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    In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl.
    Curl{v}=∇×v
    But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the gradient points along the direction of maximum increase. But what's the direction of gradient in a vector field? And why does the cross product give the Curl?
     
  2. jcsd
  3. Oct 20, 2015 #2

    Orodruin

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    It is not a cross product, it simply happens to have a similar form in Cartesian coordinates.

    You can think of (the inner product of) the curl (with a normal to the plane) as being a measure of the line integral around a small planar loop. The direction of the curl is the direction which will maximise this line integral.

    This is analogous to how you may see divergence as a measure of the net flow out of the closed surface surrounding a small volume.
     
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