we define a function $\displaystyle f$ in $\displaystyle R^n$ to have compact support if there is some $\displaystyle r > 0$ such that $\displaystyle \|\mathbf{x}\|_2 > r \implies f(\mathbf{x}) = 0$. Three questions:

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

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

$\displaystyle

\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 $\displaystyle n$ functions $\displaystyle f_i$ on $\displaystyle \mathbb{R}^n$ (none with compact support) such that

$\displaystyle

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

$

where $\displaystyle B_r$ is the ball of radius $\displaystyle r$ about the origin in $\displaystyle \mathbb{R}^n$. The coordinate functions are defined as $\displaystyle f_i(\mathbf{x}) = f_i(x_1, x_2, \cdots, x_n) = x_i$.