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Thread: Injective

  1. #1
    Aug 2009


    Let \subseteq \mathbb{R}^2" alt="f \subseteq \mathbb{R}^2" /> \to \mathbb{R}^2. Show that if the determinant of the Jacobian Matrix of f is not equal to zero at (x_o,y_o)\in D, then f is injective in some neighborhood of (x_o,y_o).
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  2. #2
    MHF Contributor

    Apr 2005
    First, the Jacobian is continuous in (x, y) (you may need to show that) so if it is not 0 at (x_0, y_0) it is not 0 in some neighborhood of that point. Now use the "mean value theorem". If f(x_1, y_1)= f(x_2, y_2) then f(x_1, y_1)- f(x_2, y_2)= J(x, y)\begin{bmatrix}x \\ y\end{bmatrix}= 0 for some point (x, y) on the line between (x_1, y_1) and (x_2, y_2)
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