Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose I have a local nbd $U$ in a manifold centered around $p$ and a chart on it. The chart is given by $X=(x_1,x_2,\cdots,x_n):U\rightarrow V\subset\mathbb{R}^n$. I consider the vector fields $\frac{\partial}{\partial x_i}$, ($i=1,\cdots,n$) which are given by the curve $c_i:(-\epsilon,\epsilon):\rightarrow U$, $c_i(t)=X^{-1}(0,0,\cdots,t,\cdots,0)$ ($t$ at the $i$-th coordinate).

I want to ask whether $\bigg[\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}\bigg]$ is always $0$ ?

share|improve this question
1  
Hint: Compute the action of your $\bigg[\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}\bigg]$ on an arbitrary smooth function $f$ –  Yuri Vyatkin Mar 1 '13 at 11:03
add comment

1 Answer 1

This is Schwarz's theorem on symmetric second derivatives.

http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives

share|improve this answer
    
I knew it as Young's theorem.. –  Berci Mar 1 '13 at 11:17
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.