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Thread: Injective

  1. #1
    Aug 2009


    Let $\displaystyle f \subseteq \mathbb{R}^2$ $\displaystyle \to \mathbb{R}^2$. Show that if the determinant of the Jacobian Matrix of $\displaystyle f$ is not equal to zero at $\displaystyle (x_o,y_o)\in D$, then $\displaystyle f$ is injective in some neighborhood of $\displaystyle (x_o,y_o)$.
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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 $\displaystyle (x_0, y_0)$ it is not 0 in some neighborhood of that point. Now use the "mean value theorem". If $\displaystyle f(x_1, y_1)= f(x_2, y_2)$ then $\displaystyle f(x_1, y_1)- f(x_2, y_2)= J(x, y)\begin{bmatrix}x \\ y\end{bmatrix}= 0$ for some point (x, y) on the line between $\displaystyle (x_1, y_1)$ and $\displaystyle (x_2, y_2)$
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