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Gradient of the scalar field

  1. Aug 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Calculate the gradient of the scalar field f(x,y) = x[tex]^{2}[/tex] - y[tex]^{2}[/tex] . Sketch the gradient for a few point on two straight lines y = x and y = -x on the plane and comment on the properties of the sketch.



    2. Relevant equations



    3. The attempt at a solution
    So I worked out the gradient to be:

    f = 2xi - 2yj

    and then I did this for the point

    x y x -y
    -2 -2 -2 2
    -1 -1 -1 1
    0 0 0 0
    1 1 1 -1
    2 2 2 -2

    but then I got confused on how to implement the gradient to this to do the sketch! Help would be much appreciated
     
  2. jcsd
  3. Aug 26, 2009 #2
    The sketch would be something like this:

    In each point on the plane, (x,y), there is a gradient vector as you said, 2xi - 2yj . these vectors point to the direction in the function's domain, which the main function has the greatest increase in its value.
     
  4. Aug 26, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    At each (x,y) draw a vector having x-component 2x and y-component -2y. That is, go to the right 2x and down 2y (assuming x and y are positive, of course).
     
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