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Homework Help: Help Inverse function theorem

  1. Dec 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Let f= (f_1, f_2, f_3) be a vector valued function defined (for every
    point (x_1,x_2,x_3) in R^3 for which x_1 + x_2 + x_3 is not equal to -1) as follows:


    f_k (x_1,x_2,x_3) = x_k /( 1+x_1+x_2+x_3) where k =1,2,3.


    After some computations I found that the determinant of the Jacobian
    matrix is (1+x_1+x_2+x_3)^(-4) (which coincides with the answer of the book).
    Then, by the inverse function theorem, it follows that f is one to one
    since the determinant is nonzero.

    The problem is the following:


    Compute f^(-1) explicitly.


    How can I do this????


    http://en.wikipedia.org/wiki/Inverse_function_theorem


    Gives a formula to find the inverse of the jacobian matrix, but I'm trying to find the inverse of the function.

    How to do this?


    3. The attempt at a solution

    I don't see how to find the inverse explicitly, I know it exists because the determinant
    of the Jacobian is nonzero everywhere.
     
  2. jcsd
  3. Dec 15, 2008 #2
    Nevermind, its just a system of equations =)
     
  4. Dec 15, 2008 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    In general it can be hard or impossible to find a formula for the inverse. In this case it's easy because it's a simple function. Just use algebra. If f(x,y,z)=(a,b,c) can you find a formula for x, y and z in terms of a, b and c? That's three simultaneous equations in the three variables if you equate the components. Hint: add them.
     
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