Trying to understand the following statement, I hope my stupid question won't bother you too much. Thank you.
Proposition. Let $M$ be an oriented smooth $n$-manifold and let $\omega$ be a compactly supported $n$-form on $M$. If $-M$ denotes $M$ with the opposite orientation, then $$\int_{-M}\omega=-\int_M\omega.$$
I'm not sure if I understand the meaning of the notation $-M$ correctly. Does it signify that each tangent space $T_p M$ is endowed with the other orientation $-\mu_p$? Here $\mu_p$ denotes the original orientation on $T_p M$. (every finite-dimensional vector space has exactly two orientations)
Thank you.