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Math Help - Sketching level curves

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    Sketching level curves

    Sketch the level curves for the function z = ln(x^2 + y^2).

    Attempt:

    k = ln(x^2 + y^2)

    z = ln(k^2 + y^2)

    z = ln(x^2 + k^2)

    Now I am not sure how I would draw each of these graphs due to the ln(x^2 + y^2) part.

    Also when asked to sketch the level curves, does that mean you just let z = k and draw one graph? (as opposed to also letting x = k and y = k)
    Last edited by SyNtHeSiS; December 9th 2011 at 05:50 AM.
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    MHF Contributor alexmahone's Avatar
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    Re: Sketching level curves

    Quote Originally Posted by SyNtHeSiS View Post
    Sketch the level curves for the function z = ln(x^2 + y^2).

    Attempt:

    k = ln(x^2 + y^2)

    z = ln(k^2 + y^2)

    z = ln(x^2 + k^2)

    Now I am not sure how I would draw each of these graphs due to the ln(x^2 + y^2) part.

    Also when asked to sketch the level curves, does that mean you just let z = k and draw one graph? (as opposed to also letting x = k and y = k)
    z=k

    \ln (x^2+y^2)=k

    x^2+y^2=e^k

    So, the level curves are circles in the xy-plane that are centered at the origin.
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    Re: Sketching level curves

    Thanks . Would drawing the graph give you a circular paraboloid (or do you have to draw the traces for x = k and y = k, before drawing x = ln(x^2 + y^2))?
    Attached Thumbnails Attached Thumbnails Sketching level curves-circular-paraboloid.jpg  
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Sketching level curves

    Quote Originally Posted by SyNtHeSiS View Post
    Thanks . Would drawing the graph give you a circular paraboloid (or do you have to draw the traces for x = k and y = k, before drawing x = ln(x^2 + y^2))?
    No, because the equation of a circular paraboloid is x^2+y^2=a^2z.
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    Re: Sketching level curves

    Oh ok. What steps would you use to draw the correct graph?
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  6. #6
    MHF Contributor alexmahone's Avatar
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    Re: Sketching level curves

    Quote Originally Posted by SyNtHeSiS View Post
    Oh ok. What steps would you use to draw the correct graph?
    z=\ln (x^2+y^2)=\ln r^2=2\ln r

    r=e^{z/2}

    I have attached a graph generated using MATLAB.
    Attached Thumbnails Attached Thumbnails Sketching level curves-pic.png  
    Last edited by alexmahone; December 16th 2011 at 03:14 AM.
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    Re: Sketching level curves

    Any function of the form z= f(x^2+ y^2)= f(r) has circular symmetry. In particular, z= ln(x^2+ y^2)= ln(r^2)= 2ln(r) is a logarithm graph. Draw that and rotate around the z-axis.
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