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Proof Involving an Implicity Defined Multivariable Function

  1. Sep 26, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that the equation G(s, t, u) = 0 implicitly defines each of the three variables s, t, and u as
    functions of the other two: s = x(t, u), t = y(s, u), and u = z(s, t). If G is differentiable and Gs,
    Gt, and Gu are all nonzero, show that

    1 = - (δu/δs) · (δs/δt) · (δt/δu)


    2. Relevant equations

    δz/δs = δz/δx · δx/δs + δz/δy · δy/δs
    δz/δt = δz/δx · δx/δt + δz/δy · δy/δt

    3. The attempt at a solution

    δG/δs = δG/δs * δs/δt + δG/δs * δs/δu
     
  2. jcsd
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