My professor demonstrated that in vector calculus that you can construct basis vectors for one, two, and three forms using the vectors $dx$, $dx$ and $dy$, as well as $dx \wedge dy$, $dy \wedge dz$, and $dx \wedge dz$...but he never explained the process thoroughly. I need help constructing a real vector space, but I don't know how.
From his assignment:
- Define the real vector space $\bigwedge^p {\bf R}^n$ for all integers $p\geq 0$. Check that your definition agrees for the cases $p=1, 2, 3$. - 1 form, 2 form, and 3 forms in vector space.
- Compute the dimension of the vector space $\bigwedge^p {\bf R}^n$.
- For a set $E\subseteq {\bf R}^n$, define the set $\Omega^p (E)$ of $p$-forms defined on $E$.
The problem is that I do not know what the omega sign and the bigwedge sign is. Could anyone please give me some hints so that I can do this by myself? I'm not hounding orders to anyone, I just need help from a different perspective.