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Area-Minimizing Surfaces

  1. Oct 12, 2007 #1
    1. The problem statement, all variables and given/known data
    "If an area-minimizing surface can be given by the graph of a function, that function satisfies the minimal surface equation:
    [tex](1+{f_y}^2)f_{xx}-2f_xf_yf_{xy}+(1+{f_x}^2)f_{yy} = 0[/tex]

    1.)Determine if the graphs of the following functions may be area minimizing:
    a.) [tex]z = 2x+4y+10[/tex]
    b.) [tex]y\tan(z)=x[/tex]
    c.) [tex]\cos(y)e^z=\cos(x)[/tex]
    d.) [tex]\sqrt{x^2+y^2}=\cosh(z)[/tex]

    2.)While it is necessary that the graph of a function satisfy the Minimal Surface Equation to be area-minimizing, it may not be sufficient. Enneper's surface satisfies the equation (you do not need to show this), but is not necessarily minimal. Find a surface with the same boundary that has less surface area."

    2. Relevant equations
    Minimal Surface Equation
    [tex](1+{f_y}^2)f_{xx}-2f_xf_yf_{xy}+(1+{f_x}^2)f_{yy} = 0[/tex]

    3. The attempt at a solution
    I already went through the calculus and found that the first 3 work with the equation, and i started working on the 4th one... and it's extremely messy. I was wondering if anybody knows a way to convert to cylindrical coordinates and use a corresponding version of the minimal surface equation? Converting the function would be easy(i think), i'm just not so sure about if it would still work.

    Also, I just don't understand the second half of the problem. I'm not sure what a surface with the same boundary is... I've seen other minimal surfaces, such as the helicoid, or the catenoid: do those work?. I also looked up Enneper's surface and... it's crazy. I read into the article and it describes it with high-degree equations, multi-variable parametrization, and different types of curvature that i've never even heard of... please help.
     
    Last edited: Oct 12, 2007
  2. jcsd
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