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In Lee's book "Introduction to Smooth Manifolds", "critical value" is defined for smooth maps between smooth manifolds with or without boundary.

But in the Proof of Sard's Theorem, the author doesn't say $\overline{V_a}$ and $B_c$ are smooth manifolds with boundary, so I think "critical value" here is defined by Jacobian Matrix.

For example, $F|_{\overline{V_a}}:\overline{V_a}\to \Bbb R^n$.

For $p\in \overline{V_a}$, if the rank of Jacobian matrix of $F|_{\overline{V_a}}$ at $p$ is less than $n$, then $p$ is a critical point of $F|_{\overline{V_a}}$, and $F|_{\overline{V_a}}(p)$ is a critical value of $F|_{\overline{V_a}}$.

Is this the author's opinion?

  • $\begingroup$ Yes, you can interpret it that way. Or you can interpret it as a critical point of $F$ that happens to lie in $\overline V_a$. They give the same result. (For $F_c$, a similar remark applies, after noting the the definition of $F_c$ actually makes sense on an open subset of $\mathbb R^{n-1}$ containing $B_c$.) $\endgroup$ – Jack Lee Jun 20 at 22:13

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