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Implicit function theorem

  1. Apr 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Assume the F(x,y,z) = 0 defines z implicitly as a function of x anf y. Show that

    2. Relevant equations

    ∂z/∂x = -(∂F/∂x)/(∂F/∂z)



    3. The attempt at a solution
    I know this question is asking about the Implicit function theorem

    So I start with F(x,y,z) =0

    define it for z = F(x,y)

    gives F(x,y,f(x,y))=0.

    My problem is where to start to show that F(x,y,f(x,y)=0 shows ∂z/∂x = -(∂F/∂x)/(∂F/∂z)

    How do I show the partial derivative ∂z/∂x of F(x,y,f(x,y)) ?
    regards
    Brendan
     
  2. jcsd
  3. Apr 5, 2009 #2

    Mark44

    Staff: Mentor

    F(x, y, f(x, y)) = 0, so, taking the partial with respect to x of both sides, and using the chain rule, you get:
    [tex]\frac{\partial F}{\partial x} \frac{\partial x}{\partial x} + \frac{\partial F}{\partial y}\frac{\partial y}{\partial x} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} = 0[/tex]

    After you simplify the left side above ([itex]\partial x/\partial x[/itex] is 1, and since x and y are independent variables in this problem, [itex]\partial y/\partial x[/itex] is 0), it's easy to show what you're asked to show.
     
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