---- > [!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 dimension equal to $\text{dim }N=\text{dim }M - \text{dim }Y$. provided it is nonempty. > > Moreover, one has $T_{p}N \subset\operatorname{ker }dF_{p}$ for all $p \in N$; by [[Rank-Nullity theorem|Rank-Nullity]], this is in fact an equality: $T_{p}N=\operatorname{ker }dF_{p}.$ ^theorem > [!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 > ```