- #1
davidge
- 553
- 21
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?
$$ \int_M (\partial_\nu w_\mu - \partial_\mu w_\nu) \ dx^\nu \wedge dx^\mu = \int_{\phi (M)} (\partial_\nu w_\mu - \partial_\mu w_\nu) \ d^2x$$
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?
