Hellow.
I don't understand why the following formula is valid.
Can you please tell me the proof?


Let $f(x,y)$ be a function on star-shaped domain of $\mathbb{R}^2$, and let $c(x,y):=\int^1\_0tf(tx,ty)dt$.
Then


$f(x,y)=\partial\_x(xc)+\partial\_y(yc)$ holds.



An integral representation of a two-variable function
$$f(x,y) = \int\_0^1 \frac{d}{dt}[t^2f(tx,ty)]\, dt$$
$$=^{\ast} 2\int\_0^1 tf(tx,ty)\, dt+x\frac{\partial}{\partial x}\int\_0^1 tf(tx,ty)\, dt+y\frac{\partial}{\partial y}\int\_0^1 tf(tx,ty)\, dt$$
$$=\partial\_x(xc)+\partial\_y(yc),\;\;\text{for}\;\;c(x,y)=\int^1\_0tf(tx,ty)\,dt.$$
In the $^\ast$ equation I used that
$df(tx,ty)/dt=(x/t)\partial f(tx,ty)/\partial x+(y/t)\partial f(tx,ty)/\partial y.$


