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Math Help - compact supports of functions and differential forms

  1. #1
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    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

    <br />
\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.<br />

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

    <br />
\lim_{r \to \infty} \int_{B_r} df_1 \wedge df_2 \wedge \cdots \wedge df_n = \infty<br />

    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.
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  2. #2
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    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.
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  3. #3
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    can sum1 plzzz advise?

    many thanks.
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