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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](http://mathoverflow.net/questions/17521/any-reference-on-multilinear-algebra). In particular, the book recommended by Georges Elencwajg in his third point could be very helpful.2011-04-23
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    Thank you for the link, but how does this relate to 1, 2, and 3 form establishment. Is it simple as substituting dx^i<...2011-04-23
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    Or is there a different format when calculating the basis vectors for 1, 2 and 3 forms?2011-04-23
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    I'm sorry, I was distracted while typing another answer. Yes, it is as easy as you said in the first comment.2011-04-23
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    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 $\mathbb{R}^n$.2011-04-23
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    Okay, so that, with the book, I can do the first two parts. Thank you for your help. But, for #3, does the book that you suggested to me shows how to define any random set (E) on R^n?2011-04-23
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    Well, you should only do this for *open* subsets $E$ of $\mathbb{R}^n$, for which the usual definition is $\Omega^{p}(E) = C^{\infty}(E) \otimes \bigwedge^{p}(\mathbb{R}^n)$.2011-04-23
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    So then, after I define the stuff above, how can I use these properties to define a wedge and differential map? for example d(dx)=02011-04-24
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    You can look this up e.g. on p.13ff of Bott and Tu, *Differential forms in algebraic topology*, Springer Graduate tets in mathematics (beginning of chapter I).2011-04-24
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    Sorry, I didn't mean to be harsh, but it is getting late here. I might add an answer to that question tomorrow, but now I'm too tired.2011-04-24
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    Actually, this [wikipedia article](http://en.wikipedia.org/wiki/Exterior_derivative) doesn't look too bad. Feel free to ask further questions in case it's too condensed.2011-04-24
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    What about a wedge map?2011-04-25
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    The all-knowing Wikipedia helps [here](http://en.wikipedia.org/wiki/Exterior_algebra), too :)2011-04-25

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