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Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by \begin{equation} f(x)\approx f(a)+Jf_a(x-a) \end{equation} where $Jf_a$ is the Jacobian Matrix at $a$.

I'd like to know the relation between $((Jf_a)^{-1})^T(x-a)$ and the function $f$. What I know is $(Jf_a)^{-1}=Jf^{-1}_{f(a)}$. Can anyone help me? Thank you in advance

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  • $\begingroup$ @Jiamprong Is it a good idea to consider the following related question: Let $\mathcal{S}$ be the vector space of all smooth maps from $\mathbb{R}^{n}$ to itself. Is there a linear map $Z:\mathcal{S} \to \mathcal{S}$ with the following property: for each $f\in \mathcal{S}$ and $a\in \mathbb{R}^{n}$ we have $(Jf_{a})^{T}=J Z(f)_{f(a)}$? $\endgroup$
    – Ali Taghavi
    Commented Feb 22, 2014 at 16:36
  • $\begingroup$ @AliTaghavi: Are they equivalent? $\endgroup$
    – Jlamprong
    Commented Feb 24, 2014 at 8:40
  • $\begingroup$ @Jiamprong They are not equivalent but I think that my question is somehow related to your question. $\endgroup$
    – Ali Taghavi
    Commented Feb 24, 2014 at 10:02

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