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If you are installing JMP or JMP Pro on a machine where one of these has. Jmp for windows 10 free download.Jmp for windows 10 free download.Jmp free download for windows.Jmp free download windows.Q:

Does a map need to be flat to be a smooth map?

Let $M$, $N$ be smooth manifolds and let $f:M \rightarrow N$ be a smooth map.
Suppose that for each $p \in M$, there exists a neighborhood $U$ of $p$ and a smooth map $\tilde f:U\rightarrow\mathbb{R}^N$ whose restriction to $U \cap f^{ -1}(p)$ is equal to $f|_{U \cap f^{ -1}(p)}$.
Does it follow that $f$ is a smooth map from $M$ to $N$?
Here is what I tried. Since $f$ is smooth, $f|_U$ is also smooth. Now, since the restriction to $U \cap f^{ -1}(p)$ is the composition $f|_U = \tilde f \circ f$, it follows that $f|_{U \cap f^{ -1}(p)}$ is smooth iff $\tilde f$ is smooth. But, clearly $\tilde f$ is smooth iff $\tilde f|_{U \cap f^{ -1}(p)}$ is smooth, and $f|_{U \cap f^{ -1}(p)}$ is smooth iff $f^{ -1}(p) \rightarrow \mathbb{R}^n$ is smooth since $f|_{U \cap f^{ -1}(p)}$ is equal to the restriction of $f$ to $U \cap f^{ -1}(p)$.
In particular, there exists neighborhoods $U_1, \dots, U_n \subset M$ of $p_1, \dots, p_n$ respectively (the collection of these neighborhoods is finite) such that the restriction of $f$ to $U_i$ is smooth for each $i$, $f$ is smooth iff the restriction of $f$ to $U_1 \cup \dots \cup U_n$ is smooth, and the

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