I was reading an article by do Carmo and Warner, which says:
"By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in the tangent plane of the hypersurface at $p$ and assigning to each point of this neighbourhood the height, with respect to the oriented normal, of the hypersurface above its tangent plane as measured in a normal coordinate system about $p$ in the ambient manifold."
I want to explicitly write such a height function. I guess we should start like this: let $M \subset \overline{M}$ the hypersurface, oriented by a unitary normal $\eta : M \to T \overline{M}$. For $p \in M$, let $\delta > 0$ be such that $\text{exp}_p : B_{\delta}(0) \subset T_p M \to B_\delta(p) \subset M$ is a diffeomorphism. If $h : B_{\delta}(0) \to \mathbb{R}$ is the referred height function, how do we compute $h(v)$ for $v \in B_{\delta}(0)$?