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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:

  1. 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.
  2. Compute the dimension of the vector space $\bigwedge^p {\bf R}^n$.
  3. 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.

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There was a related question on MathOverflow. In particular, the book recommended by Georges Elencwajg in his third point could be very helpful. – t.b. Apr 23 '11 at 19:15
    
Thank you for the link, but how does this relate to 1, 2, and 3 form establishment. Is it simple as substituting dx^i<...<dx^j for keeping track of basis vectors? – JonnyQuiznos Apr 23 '11 at 19:29
    
Or is there a different format when calculating the basis vectors for 1, 2 and 3 forms? – JonnyQuiznos Apr 23 '11 at 20:10
    
I'm sorry, I was distracted while typing another answer. Yes, it is as easy as you said in the first comment. – t.b. Apr 23 '11 at 20:11
    
I'm sorry, but I don't understand this question. The basis vectors for $1$-forms on $\mathbb{R}^3$ are $dx,dy,dz$, the ones for $2$-forms are $dx \wedge dy, dy \wedge dz, dx \wedge dz$ and for $3$-forms its $dx \wedge dy \wedge dz$. For higher dimensions you have the basis vectors $dx_{i_1} \wedge dx_{i_2} \wedge \cdots \wedge dx_{i_p}$ with $1 \leq i_1 \lt i_2 \lt \cdots \lt i_p \leq n$. for the $p$-forms on $\mathbb{R}^n$. – t.b. Apr 23 '11 at 20:40

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