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Given two smooth manifolds $M,N$ and a smooth map $F:M \to N$, it seems that the entries of the Jacobian matrix for $F$ near $a \in M$ very much should depend on the choice of coordinate charts $(\varphi, U)$ near $a$ and $(\psi,W)$ near $F(a)$, so I am confused about how the matrix should be coordinate independent.

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    $\begingroup$ The matrix depends on the chart, but the meaning of the matrix does not. Whichever chart is taken, the Jacobian is a matrix representation of the differential $dF$. $\endgroup$ – Cave Johnson Aug 19 '18 at 23:21
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    $\begingroup$ This is to be interpreted as saying that a linear mapping $V\to W$ has many different matrix representations, depending on the bases you choose for $V$ and $W$. Nevertheless, the linear map stays the same no matter what those bases are. $\endgroup$ – Ted Shifrin Aug 19 '18 at 23:45

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