# A characterization of tangent space to level set of a smooth submersion

This is corollary 5.39 in John Lee’s Introduction to Smooth Manifolds.

Corollary 5.39 Suppose $$S \subseteq M$$ is a level set of a smooth submersion $$\Phi=(\Phi^1,…\Phi^k): M \to \mathbb{R^k}$$. A vector $$v \in T_pM$$ is tangent to $$S$$ if and only if $$v\Phi^1=…=v\Phi^k=0$$.

He said that the proof is immediate. But I can’t figure it out.The author said it’s a restatement of Proposition 5.38 in a special case in which the defining function takes its value in $$\mathbb{R^k}$$.

Proposition 5.38 Suppose $$M$$ is a smooth manifold and $$S \subseteq M$$ s an embedded submanifold. If $$\Phi:U \to N$$ is any local defining function map for S, then $$T_pS = \mathrm{Ker}d\Phi_p:T_pM \to T_{\Phi(p)}N$$ for each $$p \in S\cap U$$.

The step making me stuck is the sufficiency part. Suppose that $$v\Phi^1=…=v\Phi^k=0$$, in order to use Proposition 5.38, I want to show that $$v \in \mathrm{Ker}d\Phi_p$$, but I don’t know how to reach this…The author said this proof is immediate, so may I miss something? Thanks for your help in advance.