- #1
pherytic
- 7
- 0
In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following.
$$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$
$$ \bf z = \bf r^{''} - \bf r^{'} $$
$$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z + \bf r^{'})}{|\bf r - \bf z|}$$$$\nabla\int_V d^3z \frac{1}{|\bf r - \bf z|} \int_V d^3r^{'} {f(\bf z + \bf r^{'})}$$
The book says that after the change of variables to z, it is okay to just move the 1/|r-z| term out of the r' integral. But this doesn't make sense to me, given z is still a function of r'. Am I missing something or is this mathematically incorrect?
$$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$
$$ \bf z = \bf r^{''} - \bf r^{'} $$
$$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z + \bf r^{'})}{|\bf r - \bf z|}$$$$\nabla\int_V d^3z \frac{1}{|\bf r - \bf z|} \int_V d^3r^{'} {f(\bf z + \bf r^{'})}$$
The book says that after the change of variables to z, it is okay to just move the 1/|r-z| term out of the r' integral. But this doesn't make sense to me, given z is still a function of r'. Am I missing something or is this mathematically incorrect?
