I just learned about differential forms and I'm trying to relate these ideas to vector calculus.
Let me explain what I want to do in terms of a few examples.
If I have a graph $y=x^2$ as a submanifold of $\mathbb{R}^2$ and I want to find its canonical volume form, then what I can do is find the normal vector at each point and then do a little "magic".
More precisely, let $g(x,y)=y-x^2$ then taking its gradient (or exterior derivative) we get: $$ d(y-x^2) = dy - 2x \ dx $$ Now, this is almost the volume form, except we have to swap $x$ and $y$ with $1$-forms perpendicular to them, so: $$ dx-2x\ dy $$ is a volume form on $y=x^2$. Similarly, for the $2$-sphere: $x^2+y^2+z^2=1$ in $\mathbb{R}^3$. $$ d(x^2+y^2+z^2-1) = 2x\ dx + 2y \ dy + 2z\ dz $$ Choosing the $2$-forms that are "perpendicular" to each of $dx$, $dy$ and $dz$: $$ 2x \ dy\wedge dz + 2y \ dz\wedge dx + 2z \ dx\wedge dy $$ So what am I actually doing here? I would like to know the underlying mechanism for this "magic" that I'm doing in terms of manifolds language: e.g. tangent bundles, $k$-forms, etc.