---- > [!theorem] Theorem. ([[regular value theorem]]) > Let $F:M \to Y$ be a [[smooth maps between manifolds|smooth map]] of [[smooth manifold|manifolds]]. Suppose $q \in Y$ is a [[smooth submersion|regular value]] of $F$. Then > - $N:=F ^{-1}(q)$ is either empty or an [[embedded submanifold]] of $M$ with $\text{dim }N=\text{dim }M-\text{dim }Y$ > ^theorem > [!proposition] Corollary. > For such $F$, because $N \hookrightarrow M \to Y$ > is constant and [[differential of a smooth map between smooth manifolds|pushforward]] is [[covariant functor|functorial]], $T_{p}N \subset\operatorname{ker }dF_{p}$ for all $p \in N$; [[dimension]] counting (theorem statement and [[Rank-Nullity theorem|Rank-Nullity]], this is in fact an equality: $T_{p}N=\operatorname{ker }dF_{p}.$ ^proposition > [!note] Remark. > As a consistency check: by [[Rank-Nullity theorem|RN]] $\text{dim } M= \underbrace{ \text{dim } \operatorname{ker } dF_{p} }_{ = \text{dim } T_{p}N }+ \underbrace{ \operatorname{rank}dF_{p} }_{ =\text{dim } T_{q}Y \text{ }(p \text{ is regular})}$ hence $\text{dim }N=\text{dim }M - \text{dim } Y.$ (But the proof doesn't do this to avoid circularity) ^note > [!specialization] > - [[level sets of regular values form regular surfaces]] ^specialization > [!proof]- Proof. ([[regular value theorem]]) > Skipped in our course. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```