Good Night. I am studying the Frobenius theorem. I'm reading the book Foundations of differentiable manifolds and Lie Groups; Frank Warner. In the first third part of the statement is written, "is a slice S $ Y_ {1} = 0 $", where $ Y_ {1} $ is a function of a coordinated system of coordinates. I do not understand this! Also, I do not understand the phrase "The subspace S of M with the coordinate system {$ x_ {i} | S: j = 1, ..., c\} $" which is said in the definition of Slice.
(Warner definition) 1.34 Slices Suppose that $(U,\varphi)$ is a coordinate system on $M$ with coordinate functions $x_{1} ..... x_{d}$, and that $c$ is an integer, $0\leq c \leq d$. Let $a\in \varphi(U)$, and let $ S =\{q\in U:x_{i}(q)=r_{i}(a), i = c +i .... ,d\}$. The subspace $S$ of $M$ together with the coordinate system $\{x_{j}\mid S: j= 1,..., d\}$ forms a manifold which is a submanifod of $M$ called a slice of the coordinate system $(U, \varphi)$.