What is the exterior normal to the boundary of a Riemannian manifold?

Question

Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $p \in \partial M$ and in local coordinates $(x_1,\ldots,x_n)$ near $p = (0,\ldots,0)$ the manifold $M$ is given by $\{x_n \geqslant 0\}$. What is the exterior normal to $M$ at point $p$? Is it $\bigl(-\frac{\partial}{\partial x_n}\bigr)$? Or it is a vector in the orthogonal complement to the span of $\frac{\partial}{\partial x_1}$, $\ldots$, $\frac{\partial}{\partial x_{n-1}}$ with respect to the metric $g$?

Answer 1

I would guess it is a (unit ?) vector orthogonal to $\frac{\partial}{\partial x_j}$, $j<n$ such that its scalar product with $\frac{\partial}{\partial x_n}$ is negative.

The other definition you proposed ($-\frac{\partial}{\partial x_n}$) would depend on the coordinates.