Let \(\phi\colon M\to N\) be a defining function of an embedded submanifold \(S\hookrightarrow M\). Then
\begin{equation*} T_pS=\ker d\phi_p\colon T_pM\to T_{\phi(p)}N. \end{equation*}
Remark
- If \(N=\mathbb{R}^k\) we may write \(\phi=(\phi^1,\ldots,\phi^k)\). Then \(v\in T_pS\) if it suffices for every \(i=1,\ldots,k\) \begin{equation*} v(\phi^i)=0. \end{equation*}