# compact supports of functions and differential forms

• January 7th 2010, 05:11 PM
shoplifter
compact supports of functions and differential forms
we define a function $f$ in $R^n$ to have compact support if there is some $r > 0$ such that $\|\mathbf{x}\|_2 > r \implies f(\mathbf{x}) = 0$. Three questions:

1. Show that $\int_{\mathbb{R}^n} df_1 \wedge df_2 \wedge \cdots \wedge df_n = 0$ if all the $f_i$ are continuously differentiable functions, with at least one of them having compact support.

2. prove that if $f_1$ and $f_2$ are continuously differentiable functions in $\mathbb{R}^n$ and at least one of them has compact support, then

$
\int_{\mathbb{R}^n} f_1 df_2 \wedge dx_2 \wedge \cdots \wedge dx_n = -\int_{\mathbb{R}^n} f_2 df_1 \wedge dx_2 \wedge \cdots \wedge dx_n.
$

3. Show that the coordinate functions provide an example of $n$ functions $f_i$ on $\mathbb{R}^n$ (none with compact support) such that

$
\lim_{r \to \infty} \int_{B_r} df_1 \wedge df_2 \wedge \cdots \wedge df_n = \infty
$

where $B_r$ is the ball of radius $r$ about the origin in $\mathbb{R}^n$. The coordinate functions are defined as $f_i(\mathbf{x}) = f_i(x_1, x_2, \cdots, x_n) = x_i$.
• January 7th 2010, 05:14 PM
shoplifter
i mean, clearly 2 follows from 1, just by taking the functions to be the coordinate functions from $2$ through $n$, and by taking $f_1f_2$ as the first function (this has compact support, and hence satisfies the hypotheses of 1).

can i use stokes's theorem for 1? and taking the coordinate functions for part 3 reduces to the integral being the volume of the ball $B_r$, right?

i would appreciate any comments very much.

thanks.
• January 8th 2010, 09:40 PM
shoplifter