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Visualize partial derivatives?

  1. May 24, 2009 #1
    How do you visualize a second order partial derivative with respect to x and then y?

    fxy or fyx?(same thing, but how to visualize)
     
  2. jcsd
  3. May 25, 2009 #2

    tiny-tim

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    Hi yitriana! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    You can visualise fx as a slope (on an x y z "hillside" graph), and fxx as a curvature.

    fxy would be 1/2 of the difference between the curvatures in the two "diagonal" directions, fuu and fvv where u = (x+y)/√2 and v = (x-y)/√2 :wink:
     
  4. May 25, 2009 #3

    Hurkyl

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    Another thing you might do is look at a "standard" representative.

    For example, the function f(x,y) = xy has:

    f(0,0) = 0
    fx(0,0) = 0
    fy(0,0) = 0
    fxx(0,0) = 0
    fxy(0,0) = 1
    fyy(0,0) = 0

    and all other derivatives are zero.

    So, in some sense, the behavior of this function near (0,0) sets the standard for what the effect of that derivative "looks like".
     
  5. May 26, 2009 #4
    Thank you for the replies. tiny-tim's explanation was especially helpful.
     
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