Orientation

Statements

Definition. (Citation Needed) Let $\nu$ be an 1-form on $T_p{\mathbb{R}}^m$, where $p\in M\subset {\mathbb{R}}^m$. We say that the manifold $M$ is negatively (resp. positively) oriented if the value of $\nu \left(\frac{\partial \varphi }{\partial x_1}(x_1,\dots ,x_n),\dots ,\frac{\partial \varphi }{\partial x_n}(x_1,\dots ,x_n)\right)$ is negative (resp. positive), where $\varphi (x_1,\dots ,x_n)=p$.

Remarks

An orientation is precisely some piece of information about a manifold $M$ which we can use to understand what should be sign of the Integration Of Differential Forms.

References

@Bac12, pp. 50