# What's the meaning of “critical value of $F|_{\overline{V_a}}$” and “critical value of $F_c$”?

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?

• 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 $B_c$.) – Jack Lee Jun 20 at 22:13