TITLE: why integrating only alternating forms?
QUESTION [9 upvotes]: Hello I was reviewing some concepts of differential forms. I cannot recall why only multilinear alternating forms can be integrated on manyfolds and not general multilinear forms... Why is the hypothesis of being alternating so important? In an intuitive way I'd like to understand it... Is it for having a covariant concept? Or for making some riemann sums convergent?

REPLY [3 votes]: $\def\d{\mathrm{d}}
\def\np{\star} 
\newcommand\pder[2][]{\frac{\partial #1}{\partial #2}}$Here we motivate this fact with a small example.
Suppose we wish to integrate something of the form
$\d x^1\np \d x^2$, where $\np$ is some, not necessarily alternating, product.
(The superscripts here are simply labels.)
We assume that the product is distributive and acts nicely with respect to scalar multiplication.
Under coordinate transformation we find
\begin{align*}
\d x^1\np \d x^2 
&= \left(\pder[x^1]{y^1}\d y^1+\pder[x^1]{y^2}\d y^2\right)
\np
\left(\pder[x^2]{y^1}\d y^1+\pder[x^2]{y^2}\d y^2\right) \\
&=\pder[x^1]{y^1}\pder[x^2]{y^1}\d y^1\np\d y^1 
+ \pder[x^1]{y^1}\pder[x^2]{y^2}\d y^1\np\d y^2 \\ 
& \quad + \pder[x^1]{y^2}\pder[x^2]{y^1}\d y^2\np\d y^1 
+ \pder[x^1]{y^2}\pder[x^2]{y^2}\d y^2\np\d y^2.
\end{align*}
To agree with our standard notion of integration, we must have
$$\d x^1\star\d x^2 = \det\pder[x^i]{y^j} \d y^1\np \d y^2.$$
We immediately find
\begin{align*}
\d y^1\np\d y^1 &= \d y^2\np\d y^2 = 0 \\
\d y^2\np\d y^1 &= -\d y^1\np\d y^2,
\end{align*}
that is, the product $\np$ is antisymmetric.