Could you please explain the following two questions relating to differential of a variable.
In the method of integration by parts using substitution, we have $u = f(x)$, $v = g(x)$, $du = f'(x)dx$, and $dv = g'(x)dx$. Is it just a definition to assign those values to the differentials $du$ and $dv$ or is it based on some logics?
How could the highlighted differentials $dt$ $du$ in the below text be derived? I tried to use the method in (1) above but it didn't work out as I got $sin\theta cos\theta (1 - r^2) + r(cos^2\theta - sin^2\theta)$.