In the lectures we defined a volume form on a differential manifold; in particular we explicited how to orient a $d$-dimensional submanifold of $\mathbb R^{d+c}$ (let say $M=f^{-1}(b)$, with $f\in C^\infty:\mathbb R^{d+c} \to \mathbb R^c$ and $b=(b_1,b_2,\dots,b_c)$ a regular value) using the $c$ vectors $\nabla f_i$.
First suppose that $c=1$: given the standard volume form $\omega$ on $\mathbb R^{d+1}$, the contraction $\iota (\nabla f)\omega$ is a volume form on $M$, because every vector in $TM$ is independent from $\nabla f$. Then, given a chart of $M$, say $(U, \varphi, A)$ with $A\subseteq \mathbb R^d$, we calculate $j^*\iota (\nabla f)\omega(\frac \partial {\partial \varphi_1}, \dots, \frac \partial {\partial \varphi_d})$ and if we obtain a positive number the chart is oriented.
Now comes my problem: when $c \gt 1$, we decided to use the volume form on $M$ obtained in this way: $$\omega' = \iota (\nabla f_c)\iota(\nabla f_{c-1})\dots \iota(\nabla f_1)\omega.$$ This is indeed a volume form on $M$ for the same reasons of the previous case. However suppose to change of places $f_1$ and $f_2$; so instead of $f=(f_1,f_2,\dots , f_c)^t $ now we have $f'=(f_2,f_1,\dots , f_c)^t $. The subset of points in $\mathbb R^{d+c}$ corresponding to $f'^{-1}(b')$, where $b'$ is $(b_2,b_1,\dots,b_c)$, is exactly $M$, however we obtain two opposite orientations, depending on whether we use $f$ or $f'$. The geometrical meaning of this is not too clear to me, but I have a bigger problem: suppose that I want to orient a submanifold $N$ of $S^2$, for example, and this $N$ is expressed as $g^{-1}(a)\cap S^2$, with $g\in C^\infty : \mathbb R^3\to \mathbb R$, and $a$ regular value. This implies that $g\in C^\infty (S^2)$, and so $N$ is well defined as a subvariety of $S^2$. If I consider how I defined the things, in order to orient $N$ I should use $\iota (\nabla g) \iota (\nabla h)\omega$, where $\omega $ is the standard volume form on $\mathbb R^3$ and $h:(x,y,z)^t \mapsto (x^2+y^2+z^2)$; because basically I first defined $S^2$ as a submanifold of $\mathbb R^3$ and then I defined $N$ as a submanifold of $S^2$. However I could define things in the opposite order and at the end still obtain $N$ (even if it will not be obtained as a submanifold of $S^2$); and again, I have the opposite orientation of before. What is the sense of this change of orientation? Am I missing something or things really function in this way? Thanks a lot to everyone that can clarify my ideas