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

Nonconservative gradients (not an oxymoron)

  1. Jul 15, 2009 #1

    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


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook