I Integration of forms

1. Apr 7, 2017

davidge

Yes, I know that I have already created another thread on this subject before. But, in this one, I would like to ask specifically why should we change from $M$ to $\phi (M)$ in the integral below?

$$\int_M (\partial_\nu w_\mu - \partial_\mu w_\nu) \ dx^\nu \wedge dx^\mu = \int_{\phi (M)} (\partial_\nu w_\mu - \partial_\mu w_\nu) \ d^2x$$

Why is it needed to do so? And why after doing that change, we have to substitute $dx^\nu \wedge dx^\mu$ by $d^2x$ in the integrand?

I guess this is because $M$ is a manifold while $\phi (M)$ is a function. So we need to change the integrand to something that "lives" in the function space, namely $d^2x$. Did I guess right?

2. Apr 12, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Apr 14, 2017

haushofer

This expression is wrong. On the RHS, no indices are contracted. And without any reference telling us what phi is, I don't think you'll get many reactions :)

4. Apr 14, 2017

stevendaryl

Staff Emeritus
Can you either post a link to whereever you saw this, or reproduce some of the context? I can't understand what your equality means. What is $\phi(M)$?

5. Apr 14, 2017

davidge

Thanks haushofer and stevendaryl. I got the answer to this question a while ago