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Let $f_i : \mathbb{R}^n \setminus \{0\} \to \mathbb{R}$ be a homogeneous function of degree $p \neq -1$ for every $1 \le i \le n$. Namely, $f_i(tx) = t^pf_i(x)$ for every $ x \in\mathbb{R}^n \setminus \{0\}$ and $t >0$.

We define $\omega = \sum_{i=1}^n f_i(x)dx_i$ and assume it is closed.

I want to prove that $\omega = dF$ when $F = \frac{1}{p+1} \sum_{i=1}^n x_if_i(x)$

Basically want we want to show is that $\frac{\partial F}{\partial x_i} = f_i(x)$.

By straightforward deriving I get: $$ \frac{\partial F}{\partial x_i} = \frac{1}{p+1} (\sum_{j=1}^n x_j\frac{\partial f_j(x)}{\partial x_i} + f_i(x))$$

Here I got stuck. I don't know how to continue. I also don't see how the information that $\omega$ is closed (namely, $\frac{\partial f_i(x)}{\partial x_j} = \frac{\partial f_j(x)}{\partial x_i}$ for $i \neq j$) helps me here. I also don't know how to use the fact thar $f_i$ are homogeneous because it is not given that $x_i > 0$.

Help would be appreciated.

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  • $\begingroup$ If I understand correctly, you should sum over, say, $j$ in your final formula: for instance $\partial F/\partial x_i=(p+1)^{-1}\left(\sum_j x_j \partial f_j/\partial x_i+f_i\right)$ $\endgroup$ – Brightsun Jul 2 '19 at 11:50
  • $\begingroup$ Thanks, I corrected that. $\endgroup$ – Gabi G Jul 2 '19 at 11:51
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Hint: ${d\over{dt}}_{t=1}f_i(tx)={d\over{dt}}_{t=1}t^pf_i(x)=pf_i(x)$.

You can also compute it as follows

${d\over{dt}}_{t=1}f_i(tx)=(\sum_{j=1}^{j=n}x_j{{\partial f_i}\over{\partial x_j}})_{t=1}$

$=\sum_{j=1}^{j=n}x_j{{\partial f_i}\over{\partial x_j}}$.

If you replace the second formula by the first in your computations, you obtain the result.

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  • $\begingroup$ Could you explain the first line in your answer? Why $t=1$ becomes $t=0$ and where does the $pf_i(x)$ come from? Moreover, I am not sure to which computations in my question you are referring, so if you can clear it up I would be glad. $\endgroup$ – Gabi G Jul 2 '19 at 13:17
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    $\begingroup$ @Gabi G For any positive $t$, we have $f_i(tx)=t^pf_i(x)$. Now if you take the ordinary derivative w.r. to $t$ on both sides you get $d/dt\, f_i(tx)=p t^{p-1}f_i(x)$, which is what Tsemo wrote in the first line substituting $t=1$. Then, you can also calculate the derivative using the chain rule: $d/dt\, f_i(tx)=\sum_j \partial f_i(tx)/\partial x_j\, d(tx_j)/dt=\sum_j x_j \partial f_i/\partial x_j\,$; substituting again $t=1$ yields the last formula by Tsemo. $\endgroup$ – Brightsun Jul 2 '19 at 13:41
  • $\begingroup$ @Tsemo Aristide, I edited out a $t$ from your next-to-last line, guessing it was a typo. $\endgroup$ – Brightsun Jul 2 '19 at 13:43
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    $\begingroup$ @GabiG After you establish $p f_i = \sum_j x_j \partial f_i/\partial x_j$ you can use the closedness hypothesis to conclude that $\sum_j x_j \partial f_j/\partial x_i=p f_i $ which is precisely what you need in order to conclude your argument. $\endgroup$ – Brightsun Jul 2 '19 at 13:45
  • $\begingroup$ Thanks to both of you, I get it now $\endgroup$ – Gabi G Jul 2 '19 at 13:52
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Assume $x_1$, $x_2$, $\ldots$, $x_n$ are all positive. Then, for fixed $i$, we can write $$ f_i(x_1,\ldots, x_i, \ldots, x_n) = x_i^{p} f_i\left(\tfrac{x_1}{x_i},\ldots,\tfrac{x_{i-1}}{x_i}, 1,\tfrac{x_{i+1}}{x_i}, \ldots,\tfrac{x_n}{x_i}\right)\,. $$ Now, $$ F(x)=\sum_{i=1}^n x_i f_i(x)=\sum_{i=1}^n x_i^{p+1}f_i\left(\tfrac{x_1}{x_i},\ldots,\tfrac{x_{i-1}}{x_i}, 1,\tfrac{x_{i+1}}{x_i}, \ldots,\tfrac{x_n}{x_i}\right)\,. $$ Consider, for fixed $j$, $$ \frac{\partial}{\partial x_j}F= (p+1)x_j^{p+1}f_j\left(\tfrac{x_1}{x_j},\ldots,\tfrac{x_{j-1}}{x_j}, 1,\tfrac{x_{j+1}}{x_j}, \ldots,\tfrac{x_n}{x_j}\right)\\ +\sum_{i=1}^n x_i^{p+1}\frac{\partial}{\partial x_j}f_i\left(\tfrac{x_1}{x_i},\ldots,\tfrac{x_{i-1}}{x_i}, 1,\tfrac{x_{i+1}}{x_i}, \ldots,\tfrac{x_n}{x_i}\right)\,. $$ We recognize the first term on the right-hand side as $(p+1)f_j(x)$. The second term, appropriately separating the terms in the sum, gives $$ \sum_{i\neq j} x_i^{p+1}\frac{\partial }{\partial x_j} f_i\left(\tfrac{x_1}{x_i},\ldots,\tfrac{x_{i-1}}{x_i}, 1,\tfrac{x_{i+1}}{x_i}, \ldots,\tfrac{x_n}{x_i}\right)\\ + x_j^{p+1} \frac{\partial }{\partial x_j}f_j\left(\tfrac{x_1}{x_j},\ldots,\tfrac{x_{j-1}}{x_j}, 1,\tfrac{x_{j+1}}{x_j}, \ldots,\tfrac{x_n}{x_j}\right)\\ = \sum_{i\neq j} \frac{x_i^{p+1}}{x_i}\frac{\partial f_i}{\partial x_j}\left(\tfrac{x_1}{x_i},\ldots,\tfrac{x_{i-1}}{x_i}, 1,\tfrac{x_{i+1}}{x_i}, \ldots,\tfrac{x_n}{x_i}\right)\\ +\sum_{k\neq j} \frac{-x_j^{p+1}x_k}{x_j^2} \frac{\partial f_j}{\partial x_k}\left(\tfrac{x_1}{x_j},\ldots,\tfrac{x_{j-1}}{x_j}, 1,\tfrac{x_{j+1}}{x_j}, \ldots,\tfrac{x_n}{x_j}\right)\\ =\sum_{k\neq j} x_k\left(\frac{\partial }{\partial x_j}f_k-\frac{\partial}{\partial x_k}f_j\right)=0\,. $$ In the last step we used the closedness property; note that, for $k\neq j$, $$ \frac{\partial}{\partial x_k}f_j(x)=x_j^{p-1} \frac{\partial f_j}{\partial x_k}\left(\tfrac{x_1}{x_j},\ldots,\tfrac{x_{j-1}}{x_j}, 1,\tfrac{x_{j+1}}{x_j}, \ldots,\tfrac{x_n}{x_j}\right)\,. $$ When some $x_i$ is allowed to be negative, the argument can be repeated by replacing the $1$ with $-1$ appropriately.

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