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Problem - Implicit function theorem

  1. Jun 7, 2015 #1

    ORF

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    1. The problem statement, all variables and given/known data

    I have 3 functions which define 3 constraints:
    A(x,y,z)=0 (nonlinear)
    B(y,z)=y-z=0
    C(w,x,y,z)=w-(x+y+z)=0

    I'm asked to calculate the partial of a function h, with respect to the variable w; with h=h(x,y,z).

    2. Relevant equations
    A(x,y,z)=0 (nonlinear)
    B(y,z)=y-z=0
    C(w,x,y,z)=w-(x+y+z)=0

    3. The attempt at a solution

    a. With B, make y=z in all of the equations:
    A(x,y)=0 (no lineal)
    C(w,x,y)=w-(x+2y)=0
    h=h(x,y,y)=h(x,y)

    b. The subindex point out the variable of the partial; apply the chain rule
    h_w = h_x * x_w + h_y * y_w,

    c. For the partial derivatives x_w and y_w we can use the implicit function theorem:

    x_w = - C_w / C_x = 1,
    y_w = - C_w / C_y = 1/2
    with C_w = \frac{\partial C}{\partial w}, etc

    Doubt: if I make y=z at first the result if different from the case I make y=z at the end. What case is the correct?

    Simplied example:
    h(x,y)=f(x)+g(y); the constraint is x=y; and w=x+y.

    a. Making x=y at the end
    h_w = f_x * x_w + g_y * y_w = f(x)_x + g(y)_y
    x=y,
    h_w = f(x)_x + g(x)_x

    b. Making x=y at first. The variable w=2x, h(x,x) = f(x) + g(x)
    h_w = f_x * x_w + g_x * x_w
    x_w = 1/2
    h_w = 0.5 f(x)_x + 0.5 g(x)_x

    Which is the correct?

    Thank you in advance :)

    Greetings!
     
  2. jcsd
  3. Jun 7, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    You can write three equations:
    [tex] A_x dx + A_y dy + A_z dz = 0\\
    dy - dz = 0\\
    dw - dx - dy - dz = 0
    [/tex]
    Solving for ##dx, dy, dz## in terms of ##dw## allows you to express ##dh = h_x dx + h_y dy + h_z dz## solely in terms of ##dw##.

    If you don't like using ##dv## on a variable ##v##, think instead of a finite, but small, increment ##\Delta v## and first-order Taylor expansions in terms of ##\Delta x, \, \Delta y, \, \Delta z, \, \Delta w##. You want the first-order Taylor expansion of ##h(x,y,z)## in terms of ##\Delta w##.

    Solvability of the linear system above requires a condition on ##A(x,y,z)##; that is the same condition needed for applicability of the implicit function theorem to the multivariate linear system ##F_i(x,y,z,w) = 0, i = 1,2,3##.
     
    Last edited: Jun 7, 2015
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