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Minimizing function using Calc of Variations and LaGrange Equations

  1. Jan 25, 2012 #1
    My instructor likes to explain his topics at light speed and I could barely understand how to use Calculus of variations and the La Grange equations to solve this so I need some help please.

    This is the problem:

    Consider the functional for W = w(x,y) prescribed on partial(D),

    I(W) = ∫∫√1+(Wx)^2+(Wy)^2)dxdy

    subject to the integral constraint that

    J(W) = ∫∫Wdxdy = C, where C is the constant.

    Use a Calculus of Variations approach in conjunction with the intro of a Lagrange multiplier, λ, to show that:

    λ = (Wxx(1+(Wy)^2) - 2WxWyWxy + Wyy(1+(Wx)^2)) / (1+(Wx)^2+(Wy)^2)^(3/2)

    where Wx, Wy, Wxx, Wyy, Wxy are all partials.

    Now demonstrate that:

    λ = -2/a = -2(2∏/3C)^(1/3)

    Any help at all would be great. I don't even know where to start.
     
  2. jcsd
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