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Green's Theorem and Conservative Fields

  1. Dec 19, 2009 #1
    So lets say we have the vector field x^2yi+xy^2j, obviously the field is not conservative since dq/dx-dp/dy=y^2-x^2=/=0

    however, lets say we wanted to find where locally the field would behave like a potential field, so we set y^2-x^2=0, so y=x (along the y=x line the field behaves like a conservative field). So my question is, a) is this true? b) is there some way to get an approximate scalar field whose gradiant behaves like the vector field locally along the y=x line?

    Just something I was pondering.
  2. jcsd
  3. Dec 20, 2009 #2

    Since your field is parallel to y=x line at all points, you can integrate along the line and get values of the scalar field along that line, and then generalize it somehow so that its gradient points in the direction of the field.
  4. Dec 20, 2009 #3


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    But his field isn't parallel to the y = x line. Not that I understand what you are getting at anyway...
  5. Dec 21, 2009 #4
    Locally along the y = x line it is.
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