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I'm looking at K-forms (Differential forms) and I somehow struggle a bit to understand the notation and the meaning of it. To be more precise I have problems in understanding the indexing. Let me explain what we have:

Let $U\subset \Bbb{R}^n$ be an open subset. Then a differentialform of order $k$ is defined to be an element $\omega(p)\in \bigwedge^k T_p^*(U)$ forall $p\in U$. The case where $k=1$ is clear since then $\bigwedge^1 T_p^*(U)=T_p^*(U)$ and thus we have one-forms which can be written as $$\omega(p)=\sum_{i=1}^n f_i dx_i$$ where $f_i:U\rightarrow \Bbb{R}$.Now similarly we have found a representation of differential forms of order $k$. There we can write $$\omega(p)=\sum_{i_1<...<i_k} f_{i_1...i_k}dx_{i_1}\wedge ...\wedge dx_{i_k}$$but here I'm somehow lost. I don't see why we need some index $i_j$, when does the $i$ change, and what is the meaning of $i$? It would be very helpful if someone could explain this a bit and maybe we can discuss about it.

Thanks a lot.

user123234
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  • It sounds like the confusion isn't with differential forms, but with the subscript notation. Are you able to understand the cases $k = 2$ or $k = 3$? – Jesse Madnick Jan 18 '22 at 10:50
  • I'm not sure because we have only seen examples but never "calculated" with this definitions only in the proofs which are really messi, so I understand the one-forms and why we have the representation I wrote above. I also see that there is somehow a similar way with k-forms to get to the representation, but I somehow messed up with all the indexes. So I don't think that I understand the cases $k=2,3$ only by looking at the theory. – user123234 Jan 18 '22 at 10:58

1 Answers1

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For concreteness, let's take $n = 4$, so the coordinates on $\mathbb{R}^4$ are $(x_1, x_2, x_3, x_4)$.

$\bullet$ A differential $1$-form $\omega$ on an open set $U \subset \mathbb{R}^4$ can be written $$\omega = \sum_{1 \leq i \leq 4} f_i\,dx_i = f_1dx_1 + f_2\,dx_2 + f_3\,dx_3 + f_4dx_4$$ for some functions $f_1, \ldots, f_4 \colon U \to \mathbb{R}$.

$\bullet$ A differential $2$-form $\omega$ on an open set $U \subset \mathbb{R}^4$ can be written \begin{align*} \omega & = \sum_{1 \leq i < j \leq 4} f_{ij}\,dx_i \wedge dx_j \\ & = f_{12}\,dx_1 \wedge dx_2 + f_{13}\,dx_1 \wedge dx_3 + f_{14}\,dx_1 \wedge dx_4 \\ & \ \ \ \ \ \ \ + f_{23}\,dx_2 \wedge dx_3 + f_{24} \,dx_2 \wedge dx_4 + f_{34}\,dx_3 \wedge dx_4 \end{align*} for some functions $f_{12}, f_{13}, f_{14}, f_{23}, f_{24}, f_{34} \colon U \to \mathbb{R}$.

$\bullet$ A differential $3$-form $\omega$ on an open set $U \subset \mathbb{R}^4$ can be written \begin{align*} \omega & = \sum_{1 \leq i < j < m \leq 4} f_{ijm}\,dx_i \wedge dx_j \wedge dx_{m} \\ & = f_{123}\,dx_1 \wedge dx_2 \wedge dx_3 + f_{124}\,dx_1 \wedge dx_2 \wedge dx_4 \\ & \ \ \ \ \ + f_{134}\,dx_1 \wedge dx_3 \wedge dx_4 + f_{234}\,dx_2 \wedge dx_3 \wedge dx_4 \end{align*} for some functions $f_{123}, f_{124}, f_{134}, f_{234} \colon U \to \mathbb{R}$.

$\bullet$ A differential $4$-form $\omega$ on an open set $U \subset \mathbb{R}^4$ can be written \begin{align*} \omega & = \sum_{1 \leq i < j < m < p \leq 4} f_{ijmp}\,dx_i \wedge dx_j \wedge dx_{m} \wedge dx_p = f_{1234} \,dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4 \end{align*} for some function $f_{1234} \colon U \to \mathbb{R}$.

In general, when we work with differential $k$-forms on $\mathbb{R}^n$, we have to resort to writing the summation range as $$1 \leq i_1 < i_2 < \cdots < i_k \leq n$$ for the simple reason that we don't have infinitely many letters in the alphabet.

Jesse Madnick
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  • oh wow thanks a lot. But for example in the differential 3 form what are my $i_1...$ – user123234 Jan 18 '22 at 11:05
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    For the case of $3$-forms in $\mathbb{R}^4$, I renamed $(i_1, i_2, i_3)$ to be $(i,j,m)$, because I thought that might make the pattern a little clearer. Similarly, in the case of $2$-forms in $\mathbb{R}^4$, I renamed $(i_1, i_2)$ to be $(i,j)$. – Jesse Madnick Jan 18 '22 at 11:06
  • aha so our $i$ has no explicit meaning for example $1\leq i\leq q$ for some $q$ – user123234 Jan 18 '22 at 11:07
  • Yes, the subscripts $(i_1, \ldots, i_k)$ are dummy variables; they're simply placeholders to indicate the summation. You could call them anything you want. The caveat is that they have to satisfy $1 \leq i_1 < i_2 < \cdots < i_k \leq n$. – Jesse Madnick Jan 18 '22 at 11:09
  • ah so now it makes much more sense, I was always wondering about these $i$'s. Sorry for another stupid question, is there also a way to think geometrically about this differential forms i.e. for example these in your example? – user123234 Jan 18 '22 at 11:11
  • Yes, but that's a separate question. The geometric meaning of differential forms has been discussed several times on this site, for example here: https://math.stackexchange.com/questions/440816/geometric-understanding-of-differential-forms/2596783#2596783 I especially like Jules' answer on that thread. It is also discussed on this other thread: https://math.stackexchange.com/questions/548131/whats-the-geometrical-intuition-behind-differential-forms?noredirect=1&lq=1 – Jesse Madnick Jan 18 '22 at 11:13
  • perfect thanks so I look around there! Thanks for your help. – user123234 Jan 18 '22 at 11:14
  • sorry again a short question $f_{12}$ in your example above is this one explicit function even though it has two indices, right? – user123234 Jan 18 '22 at 11:18
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    Yes, $f_{12}(x_1,x_2,x_3,x_4)$ is a single function of the $4$ variables $(x_1, x_2, x_3, x_4)$, just like $f_{234}$ is a single function, etc. So, for example, $f_{12}$ could be $f_{12}(x_1, x_2, x_3, x_4) = \sin(x_1 + x_2 - x_3)\cos(x_4)$ or something like that. – Jesse Madnick Jan 18 '22 at 11:19
  • perfect now it's clear thanks – user123234 Jan 18 '22 at 11:20