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.

I know that $X = \sum_i a_i \frac \partial\partial_{x_i}$ and $Y=\sum_j b_j \frac \partial\partial_{x_j}$ but I'm not sure how to proceed. The only approach I can think of is something to do with the chain rule.

share|improve this question
    
How do you understand $XY$? –  janmarqz Feb 9 at 0:40
    
Is $XY$ the product of the summations that I gave above? –  user127292 Feb 9 at 1:22
    
not really... what it is $XY$ is to derivate $Y$ in direction $X$. Example: $\frac{\partial}{\partial v}(v^2w\frac{\partial}{\partial w})=2vw\frac{\partial}{\partial v}+v^2w\frac{\partial}{\partial v}\frac{\partial}{\partial w}$ –  janmarqz Feb 9 at 1:28
    
so Xg is the derivative to g in the direction X? Then what does gY mean? –  user127292 Feb 9 at 1:31
    
$gY=\sum_j gb_j \frac \partial\partial_{x_j}$ –  janmarqz Feb 9 at 1:32

1 Answer 1

up vote 0 down vote accepted

Hint:

$$XY=\sum_i a_i \frac{\partial}{\partial{x_i}}\left(\sum_j b_j\frac {\partial}{\partial x_j}\right),$$

$$=\sum_{ij} a_i \frac{\partial}{\partial{x_i}}\left(b_j \frac {\partial}{\partial{x_j}}\right),$$

$$=\sum_{ij} a_i \left(\frac{\partial b_j}{\partial{x_i}}\frac{ \partial}{\partial{x_j}} +b_j\frac{\partial}{\partial{x_i}}\frac{\partial}{\partial{x_j}}\right),$$

$$=\sum_{ij} a_i \left(\frac{\partial b_j}{\partial{x_i}}\frac{\partial}{\partial{x_j}}+b_j\frac{\partial}{\partial{x_i}}\frac{ \partial}{\partial{x_j}}\right),$$

$$=\sum_{ij} a_i\frac{\partial b_j}{ \partial{x_i}}\frac{ \partial}{\partial{x_j}}+\sum_{ij} a_ib_j\frac{\partial}{\partial{x_i}}\frac{ \partial}{\partial{x_j}}.$$

share|improve this answer

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.