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Jacobian Determinant/ mult. variable implicit differentiation

  1. Jul 16, 2013 #1
    1. The problem statement, all variables and given/known data
    Let F: x^2 + y^2 - z^2 + 2xy - 1 = 0 and G: x^3 + y^3 - 5y - 4 = 0. Calculate dz/dx. Note: This is NOT the partial derivative ∂z/∂x.

    I do not need help in taking the derivative of many polynomials. What I need help in is setting up a Jacobian determinant to evaluate this.

    2. Relevant equations

    The Jacobian determinant, bit tough to explain in a short period of space. Exceptionally helpful with many equations with as many variables as equations, in which the partial derivative can be evaluated by the Jacobian Determinant.

    3. The attempt at a solution
    Okay, so dz/dx = [∂(F, G) /∂(z, y)] / ∂(F, G) / ∂(x, y) * (-1), but this is incorrect as this is ∂z/∂y. Essentially I need to know, what is a function of what, and how can I evaluate this? I do not care necessarily for the answer, simply for the setup.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jul 17, 2013 #2


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    Homework Helper

    The ultimate independent variable is x. G relates x and y so y is only function of x:y=y(x).
    F is a relation among x, y, z. z depends both on y and x: z=z(x,y(x)).

  4. Jul 17, 2013 #3


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    Homework Helper

    Here is the motivation
    0=dF=Fx dx+Fy dy+Fz dz
    0=dG=Gx dx+Gy dy+Gz dz
    first we find
    0=(Gy dF - Fy dG)/dx
    which will have the Jacobians you seek
    then think it through in general

    What happens is you get F,G,x,y in one Jacobian and F,G,y,z in the other and F,G,y effectively cancel (chain rule) leaving dz/dx
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