Toronto Math Forum
APM3462016F => APM346Lectures => Chapter 6 => Topic started by: Shentao YANG on November 05, 2016, 06:51:32 PM

Can any one explain in detail to me how we get these two formula in section 6.3:
http://www.math.toronto.edu/courses/apm346h1/20169/PDEtextbook/Chapter6/S6.3.html
$$\int\!\!\!\int \Delta u \cdot v\,dxdy =  \int\!\!\!\int \nabla u \cdot \nabla v\,dxdy$$
$$\int\!\!\!\int\!\!\!\int
\Delta u \cdot v{\rho ^2}\sin (\phi )\,d\rho d\phi d\theta =  \int\!\!\!\int\!\!\!\int
( {u_\rho }{v_\rho } + {1 \over {{\rho ^2}}}{u_\phi }{v_\phi } + {1 \over {{\rho ^2}\sin (\phi )}}{u_\theta }{v_\theta }){\rho ^2}\sin (\phi )\,d\rho d\phi d\theta = \int\!\!\!\int\!\!\!\int
( {({\rho ^2}\sin (\phi ){u_\rho })_\rho } + {(\sin (\phi ){u_\phi })_\phi } + {({1 \over {\sin (\phi )}}{u_\theta })_\theta })v\,d\rho d\phi d\theta .$$
By the way, I think the equation $(6)'$ in
http://www.math.toronto.edu/courses/apm346h1/20169/PDEtextbook/Chapter6/S6.3.html#mjxeqneq6.3.6
is wrong, I guess the denominator of the last term should be ${\rho ^2}{\sin ^2}(\varphi )$ instead of ${\rho ^2}{\sin}(\varphi )$

The first equality is due to integration by parts and Gauss formula. The second equation is the first one rewritten in spherical coordinates. Please check again if there is any misprint

Misprint at where? I guess the link is pointing to a wrong equation, equation $(6)$ is correct, but $(6)'$, I guess, leave out a square.

Indeed, there is a square in (6)'.