I have a question.

Question

I have a question.

anonymous · Answer

If $R(x)$ is a $C^\infty$-function near the origin in $\mathbb{R}^n$ satisfying $R(0)=0$ and $DR(0)=0$, show that there exist smooth functions $r_{jk}(x)$ such that$$R(x)=\sum r_{jk}(x)x_jx_k.$$I know that, by the fundamental theorem of calculus, applied to $\phi(t)=F(x+ty)$,$$F(x+y)=F(x)+\int_0^1DF(x+ty)y\text{ }dt,$$provided $F$ is $C^1$, we can write $R(x)=\Phi(x)x$, $\Phi(x)=\int_0^1DR(tx)dt$, since $R(0)=0$. Then $\Phi(0)=DR(0)=0$, so we can apply the above theorem again, to give $\Phi(x)=\Psi(x)x$. I get stuck here, however. Do you guys have suggestions on how to continue?

turingtest · Answer

Do you ever try mathstackexchange for your questions?
In general I think they are more of that websites caliber