# Toronto Math Forum

## APM346-2016F => APM346--Lectures => Chapter 6 => Topic started by: Shentao YANG on November 05, 2016, 06:51:32 PM

Title: Question about the deviation of laplacian
Post 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/PDE-textbook/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/PDE-textbook/Chapter6/S6.3.html#mjx-eqn-eq-6.3.6
is wrong, I guess the denominator of the last term should be ${\rho ^2}{\sin ^2}(\varphi )$ instead of ${\rho ^2}{\sin}(\varphi )$
Title: Re: Question about the deviation of laplacian
Post by: Victor Ivrii on November 05, 2016, 09:21:20 PM
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
Title: Re: Question about the deviation of laplacian
Post by: Shentao YANG on November 05, 2016, 11:10:46 PM
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.
Title: Re: Question about the deviation of laplacian
Post by: Victor Ivrii on November 06, 2016, 12:05:43 AM
Indeed, there is a square in (6)'.