While studying manifolds I am having some problem with definition of manifold with boundary.Let $S$ be a regular $n$-level surface in $\mathbb R^{n+1}$ with boundary defined by $S=f^{-1}(0)\cap (\bigcap\limits_{i=1}^k g_i^{-1}(-\infty,c_i]$ where $\nabla f(p)\neq 0$ for all $p\in S$ and $\{\nabla f(x),\nabla g_i(x)\}$ are linearly independent of all $x\in S\cap g_i^{-1}(c_i)$ and $S\cap g_i^{-1}(c_i)\cap g_j^{-1}(c_j)=\phi$.
The defintion of tangent space of the regular level surface with boundary is defined to be $T_pS=\{v\in T_p\mathbb R^{n+1}:\nabla f(p).v=0\}$
Now I am a little confused with the tangent space to the boundary.
Our instructor had defined it as follows:
If $S$ is as above and $p\in \partial S$ then suppose $p\in g_i^{-1}(c_i)$ and let $v\in T_pS$.Then the tangent space to the boundary $\partial S$ is defined as $\{v\in T_pS:v$ is orthogonal to the boundary$\}=T_p(\partial S)$.But it is not clear what it means when we say orthogonal to the boundary and $v$ is called normal the boundary if $v.w=0$ for all $w\in T_p(\partial S)$.I am looking for a clear definition of what is the rigorous definition of tangent space to the boundary manifold and the definition of normal to the boundary.Our instructor also made a mistake that $\nabla g_i$ is always perpendicular to $\nabla f$ which is not true.So,all the definitions got a bit confusing.Can someone help me resolve this issue?