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Homework Help: Another one on contours

  1. Feb 21, 2006 #1
    I shall plot the contour diagram of

    [tex] z = (x^2 - y^2) * e^ {-x^2 - y^2} [/tex]

    for z = O this is easy, however, if z = 1 one gets

    [tex] ln (x^2-y^2) = x^2 + y^2 [/tex]

    Does anyone know how to draw this?

    I tried to find the intersection between two functions y1 and y2 being the lhs and rhs of the above equation respectively; but since I don't know how to draw y = ln (x^2 - y^2) either I have no clue how this is supposed to work.
     
  2. jcsd
  3. Feb 21, 2006 #2

    AKG

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    Maybe start by looking at x² as X, and y² as Y, so you're just looking at ln(X-Y) = X+Y. Also, you do not want to see where y = ln(x² - y²) and y = x² + y² intersect. For example, if you wanted to plot the diagram for

    cos(y) = sin(y)

    you wouldn't want to see where:

    y = sin(y), and y = cos(y)

    intersected. y = cos(y) will just be the horizontal line y = 0.74 (approximately), and y = sin(y) will just be the horizontal line y = 0, so their intersection will be empty. On the other hand, sin(y) = cos(y) will have a solution consisting of infinitely many horizontal lines, a distance of [itex]\pi[/itex] apart from one another.

    So you really just have to solve ln(X-Y) = X+Y.
     
  4. Feb 21, 2006 #3
    thanks for this. I do understand your argument. however, I do not understand how I could draw ln(X-Y) = X+Y either, since solving for x and y looks impossible.


     
  5. Feb 22, 2006 #4

    saltydog

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    I belive you're approaching it wrong. Do not take the log but rather solve for y in the equation:

    [tex]a=(x^2-y^2)e^{-(x^2+y^2)}[/tex]

    Yea, it could happen via the Lambert W function. How about I start it for you:

    Switch it around to:

    [tex](x^2-y^2)=ae^{(x^2-y^2)}[/tex]

    Now here comes the tricky part: What do I have to multiply both sides by so that the LHS is:

    [tex]fe^{f}[/tex]

    (where f is some algebraic expression)

    That's the form to extract the Lambert-W function. Then solve for y . . . equal rights and all that stuf.:smile:
     
    Last edited: Feb 22, 2006
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