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Nonconservative gradients (not an oxymoron)

  1. Jul 15, 2009 #1
    fx,fy=-y,x
    curl(fx,fy)=2

    its counterclockwise so the gradient is pointing ccw.
    the gradient is the direction of max increase, so
    then the surface z=f[x,y] whose gradient is -y,x is a spiral/screw whose z position goes up forever as x and y are traced out counterclockwise

    so, a nonconservative field IS a gradient of a surface.
    why was i lied to?
     
  2. jcsd
  3. Jul 15, 2009 #2

    Office_Shredder

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    I think you're going to need to write this out clearer. How can the curl of a vector be a number? How can the gradient of a three variable equation be a two dimensional vector?
     
    Last edited: Jul 15, 2009
  4. Jul 16, 2009 #3
    Hello there! Okay, lets take it step by step and yes, please do be a little clearer so we can understand exactly what you're saying. I think what you're saying is that you're trying to find the surface whose gradient at each point is (-y,x,0)? How did you make these out to be spirals?


    So let's see, you've got the vector field:

    [tex]\vec{V} = (-y, x, 0)[/tex]

    Ye? And you've calculated its curl, which is:

    [tex]Curl \vec{V} = (0,0,2)[/tex]

    so the field is 'rotating' counterclockwise viewed from above (the positive z axis). The curl of your vector field is constant in all of space, and non-zero. Because it is non-zero means that your vector field [tex]\vec{V} = (-y, x, 0)[/tex] cannot be a gradient, in other words, there is no scalar field whose gradient is [tex]\vec{V}[/tex]. That is, there are no surfaces whose gradient at each point is (-y, x, 0). But! this does not mean that there are no surfaces which are orthogonal at each point to [tex]\vec{V}[/tex]. In fact, in your case, these surfaces DO exist, because the vector field [tex]\vec{V}[/tex] is normal to its curl.
     
  5. Jul 16, 2009 #4
    Just to help visualize, here is the curl of your vector field V, as viewed from above:

    http://img31.imageshack.us/img31/4605/curl.gif [Broken]
     
    Last edited by a moderator: May 4, 2017
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