**Definition 6.3.1 (Orientation of vector space).**Let $V$ be a finite-dimensional real vector space, and let $\mathcal{B}_V$ be the set of bases of $V$. An orientation of $V$ is a map $\Omega: \mathcal{B}V \to \{+1, -1\}$ such that if $\{v\}$ and $\{v'\}$ are two bases with change of basis matrix $[P{v' \to v}]$, then

$$ \Omega(\{v'\}) = \text{sgn}(\det [P_{v' \to v]}) \Omega(\{v\}). $$

A basis $\{w\} \in \mathcal{B}_V$ is called direct if $\Omega(\{w\}) = +1$. It is indirect if $\Omega(\{w\}) = -1$.

Definition 6.3.3 (Orientation of manifold). An orientation of a $k$-dimensional manifold $M\subset \R^n$ is a continuous map $\mathcal{B}(M) \to \{+1, -1\}$ whose restriction to each $\mathcal{B}_xM$ is an orientation of $T_xM$.

Proposition 6.3.9 (Orienting manifolds given by equations). Let $U \subset \R^n$ be open, and let $f: U \to \R^{n-k}$ be a map of class $C^1$ such that $[Df(x)]$ is surjective at all $x\in M = f^{-1}(0)$. Then the map $\Omega_x: \mathcal{B}(T_xM) \to \{+1, -1\}$ given by

$$ \Omega(v_1, \dots, v_k) = \text{sgn}\det [\nabla f_1(x), \dots, \nabla f_{n-k}(x), v_1, \dots, v_k] $$

is an orientation of $M$.

Proposition 6.3.10 (Orientation of connected, orientable manifold). If $M$ is a connected manifold, then either $M$ is not orientable, or it has two orientations. If $M$ is orientable, then specifying an orienation of $T_xM$ at one point defines the orientation at every point.