Definition 6.4.1. (Orientation-preserving linear transformation). Let $V$ be a $k$-dimensional vector space oriented by $\Omega: \mathcal{B}(V) \to \{+1, -1\}$. A linear transformation $T: \R^k \to V$ is orientation preserving if

$$ \Omega (T(\vec e_1) ,\dots, T(\vec e_k)) =+1. $$

It is orientation reversing if

$$ \Omega(T(\vec e_1), \dots, T(\vec e_k)) = -1. $$

Note that if $T$ is not invertible, then it is neither because it is not a basis of $V$.

Definition 6.4.2 (Orientation-preserving parametrization of a manifold). Let $M\subset \R^m$ be a $k$-dimensional manifold orientaed by $\Omega$, and let $U \subset \R^k$ be a subset with boundary of $k$-dimensional volume 0. Let $\gamma: U \to \R^m$ parametrize $M$ as described in Definition 5.2.3, so that the set $X$ of trouble spots satisfies all conditions of that definition; in particular, $X$ has $k$-dimensional volume 0. Then $\gamma$ is orientation preserving if for all $u \in (U-X)$, we have

$$ \Omega(D_1\gamma(u), \dots, D_k\gamma(u)) = +1. $$

Note that the vectors form a basis of $M$'s tangent space at point $u$. $T_{\gamma(u)}M$.

Proposition 6.4.6 (Checking orientation at a single point). Let $M$ be an oriented manifold, and let $\gamma: U \to M$ be a parametrization of an open subset of $M$, with $U-X$ connected, where $X$ is as in Definition 5.2.3. Then if $\gamma$ preserves orientation at as single point of $U$, it preserves orientation at every point of $U$.

Proposition 6.4.8 (Orientation-preserving parametrization). Let $M\subset \R^n$ be a $k$-dimensional oriented manifold. Let $U_1, U_2$ be subsets of $\R^k$, and let $\gamma_1: U_1 \to \R^n$ and $\gamma_2 : U_2 \to \R^n$ be two parametrizations of $M$. Then $\gamma_1$ and $\gamma_2$ are eitehr both orientation preserving or both orientation reversing if and only if for all $u_1\in U_1^{OK}$ and $u_2\in U_2^{OK}$ with $\gamma_1(u) = \gamma_2(u)$ we have

$$ \det [D(\gamma_2^{-1}\circ\gamma_1)(u)] > 0. $$

Theorem 6.4.10 (Integral independent of orientation-preserving parametrization). Let $M\subset \R^n$ be a $k$-dimensional oriented manifold, and let $U_1, U_2$ be two open subsets of $\R^k$, and let $\gamma_1: U_1\to \R^n$ and $\gamma_2: U_2 \to \R^n$ be two orientation-preserving parametrizations of $M$. Then for any $k$-form $\varphi$ defined on a neighborhood of $M$,

$$ \int_{[\gamma_1(U_1)]}\varphi = \int_{[\gamma_2(U_2)]} \varphi. $$

If $\gamma_1$ and $\gamma_2$ are both orientation reversing, then the integrals are equal. If one is preserving and teh other is reversing, then

$$ \int_{[\gamma_1(U_1)]}\varphi =- \int_{[\gamma_2(U_2)]} \varphi. $$

If $\det [D(\gamma_2^{-1}\circ \gamma_1)]$ is positive in some regions and negative in others, then the integrals are probably unrelated.