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Implicit Function Theorem

  1. Jun 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Can the equation x^2 + y^2 + z^2 = 3, xy + tz = 2, xz + ty + e^t = 0 be solved for x, y, z as C^1 functions of t near (x, y, z, t) = (-1, -2, 1, 0)?

    2. Relevant equations



    3. The attempt at a solution

    The mixed-partial derivatives matrix I got was:
    [2x, 2y, 2z, 0]
    [y, x, t, z]
    [z,t,x, y+e^t]

    Plugging in the numbers I get:
    [-2, -4, 2, 0]
    [-2, -1, 0, 1]
    [1, 0, -1, -1]

    I know the theorem states that when this matrix is invertible, it means explicit functions exist, however, how should I proceed as the matrix is not square?
     
  2. jcsd
  3. Jun 29, 2009 #2

    Mark44

    Staff: Mentor

    The point in question, (-1, -2, 1, 0) is not on the sphere x2 + y2 + z2 = 3. IOW, the first three coordinates of this point don't satisfy the sphere's equation. That seems problematic to me.
     
  4. Jun 29, 2009 #3
    I just looked up the errata for the textbook I was using. It looks like the author made a mistake and the first equation is actually supposed to be x^2 + y^2 + z^2 = 6.

    So with that, and re-reading the theorem, I think the answer should be as follows:

    The mix-partial derivatives matrix of the variables I want to solve for is:
    [-2, -4, 2]
    [-2, -1, 0]
    [1, 0, -1]

    Determinant of this is non-zero. Therefore, the conclusion is that explicit functions are possible.
     
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