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I want to understand the concept of $p$-forms, but I have yet trouble to relate this concept to things I already know/understand.

Def.: $U \subset \mathbb{R}^n$ be open. $ w : U \times \mathbb{R}^n \times \dots \mathbb{R}^n \to\mathbb{R}$ continuous. $w$ is called alternating, $w(x; h_1;\dots; h_i; \dots ; h_j ; \dots ; h_p) = -w(x; h_1; \dots ; h_j ; \dots ; h_i; \dots ; h_p). $ Is $w$ alternating and linear in $h_1; \dots ; h_p$ so $w$ is called a $p$-form.

I would start out with the $1$-forms as they seem to be the most accessible, so I have:

$w(x,h) = w(x, (h_1, \dots, h_n)) = \sum\limits_{i = 1}^{n} m_i *w(x, e_i) = \sum\limits_{i = 1}^{n} a_i(x) m_i $ $\ \ \ \ $ $(*) $

with $ m_i := h_i(e_i), a_i := w(x,e_i) $ with $e_i$ denoting the i'th canonical unit vector of $\mathbb{R}^n$, this is something I can work with.

Now many book author or lecturers define $dx_i$ as $ m_i $, however I have already seen $dx_i$ before in Calc 1 and 2 as a mere tool/Symbol. At that point it also didn't mean a lot, similar to this case: What does $dx$ mean?

However here $dx_i$ is now used as an extra function and in my opinion something totally different. $dx$ may still denote something like the "slope" or derivative $m$ of a linear function $f(x) = m*x$ (and easily explain why $d(dx_i) = 0$ as $f''(x) = 0$ ), but I find one has actually lost more information than gained.

(A move backwards to Riemann Integrals, rather than forward to a generalization for the sake of increasing readability and allowing to compute the length of curves in $U$) This definition however seems to me rather forced, is there a more intuitive way of thinking about the curve + the $h_i$ for example?Why uses someone this notation?

For the general $p$ -forms I would like to write it in a similar manner to $(*)$.

For $n = 2$ and $p = 2$, I get: $w(x,h, g) = w(x, (h_1,h_2), (g_1, g_2)) = a(x) (h_1 *g_1 +h_1 *g_2 + h_2*g_1 + h_2 *g_2) \ \ \ \ (\star \star) $

However I am supposed to get: $w(x,h,g) = a(x) *det \begin{pmatrix} h_1 & g_1 \\ h_2 & g_2 \\ \end{pmatrix} = a(x) \ dx_1 \wedge dx_2 $

Here I am a bit off and wonder, what my mistake is, going back to the definition and using $w$ is alternating didn't solve my problem in $ (\star \star) $.

I am very sorry my post has become so lengthy, but also didn't know how to shorten. I guess this is always the case, if one is confused / doesn't understand much. Thus I would be very happy, if someone can follow my thoughts and clear things up. As always thanks in advance for any constructive comment, answer or recommendation for further studying.

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Imago
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    I suggest you a lecture of: http://www.math.ucla.edu/~tao/preprints/forms.pdf – Emilio Novati Mar 03 '16 at 17:01
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    A comment about learning strategy: If you're really looking to "reduce this concept to things your already know/understand," you're doomed to failure. You'll be much better off if you aim to relate the concept to things you understand, while being aware that it's a new thing, not reducible to anything you already know. $p$-forms are related to line and surface integrals, cross products, divergence, gradient, curl, the right-hand rule, determinants, Jacobians, volumes, differential equations, Hamiltonian systems, and many other things; but they're more than the sum of those parts. – Jack Lee Mar 03 '16 at 20:49
  • Oh yes, you are absolutely right and agree with you. I wasn't sure, which verb to use in the beginning and seemed to have picked a bad one. Nonetheless my motivation is to always start testing out these "new rules" I get, when I read about a new chapter/topic I know yet very little or nothing about. It's sort of like experimenting and curiosity driven. – Imago Mar 03 '16 at 21:09

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