Toronto Math Forum

APM346-2016F => APM346--Lectures => Chapter 6 => Topic started by: Luyu CEN on November 04, 2016, 01:20:01 AM

Title: Change of Variable
Post by: Luyu CEN on November 04, 2016, 01:20:01 AM

\begin{equation*}x=r\cos (\theta)\end{equation*}, \begin{equation*}y=r\sin(\theta)\end{equation*}
\begin{equation*}
\left\{
\begin{aligned}
&r=\sqrt{x^2+y^2},\\
&\theta = \arctan \bigl(\frac{y}{x}\bigr);
\end{aligned}\right.
\end{equation*}

\begin{equation}
\left\{\begin{aligned}
&r_x=\frac{x}{\sqrt{x^2+y^2}}=\frac{r\cos(\theta)}{r}=\cos(\theta),\\ 
&r_y=\frac{y}{\sqrt{x^2+y^2}}=\frac{r\sin(\theta)}{r}=\sin(\theta),\\ 
&\theta_x = \frac{1}{(\frac{y}{x})^2+1} \cdot -\frac{y}{x^2}=-\frac{y}{y^2+x^2}=-\frac{r\sin(\theta)}{r^2}=-r^{-1}\sin(\theta),\\ 
&\theta_y = \frac{1}{(\frac{y}{x})^2+1} \cdot \frac{1}{x}=\frac{x}{y^2+x^2}=\frac{r\cos(\theta)}{r^2}=r^{-1}\cos(\theta).
\end{aligned}\right.
\end{equation}
By the chain rule
\begin{equation}
\left\{
\begin{aligned}
&\partial_x = r_x\partial_r  + \theta_x\partial_\theta  =\cos(\theta)\partial_r - r^{-1}\sin(\theta)\partial_\theta,\\
&\partial_y = r_y\partial_r  + \theta_y\partial_\theta  = \sin(\theta)\partial_r + r^{-1}\cos(\theta)\partial_\theta
\end{aligned}\right.
\end{equation}
Plug the above into the laplacian and expand the squares
\begin{multline*}
\Delta = \partial_x^2+ \partial_y^2=
\bigl(\cos(\theta)\partial_r - r^{-1}\sin(\theta)\partial_\theta\bigr)^2 +
\bigl(\sin(\theta)\partial_r +
r^{-1}\cos(\theta)\partial_\theta\bigr)^2 =\\
\bigl(\cos(\theta)\partial_r\cos(\theta)\partial_r -\cos(\theta)\partial_r\cdot r^{-1}\sin(\theta)\partial_\theta -r^{-1}\sin(\theta)\partial_\theta \cos(\theta)\partial_r+r^{-1}\sin(\theta)\partial_\theta r^{-1}\sin(\theta)\partial_\theta\bigr) + \\
\bigl(\sin(\theta)\partial_r\sin(\theta)\partial_r+ \sin(\theta)\partial_r r^{-1}\cos(\theta)\partial_\theta + r^{-1} \cos(\theta) \partial_\theta \sin(\theta)\partial_r+r^{-1}\cos(\theta)\partial_\theta r^{-1}\cos(\theta)\partial_\theta\bigr).
\end{multline*}
By the product rule, we have
\begin{multline*}
\Delta=
\bigl(\cos^2(\theta)\partial_r^2 + \frac{1}{r^2}\cos(\theta)\sin(\theta)\partial_\theta +r^{-1}\sin^2(\theta)\partial_r-2r^{-1}\sin{\theta}\cos{\theta}\partial_r\partial_\theta+ r^{-2}\cos^2(\theta)\partial_\theta^2 - r^{-2}\cos(\theta)\sin(\theta)\partial_\theta\bigr) +\\
\bigl(\sin^2(\theta)\partial_r^2 - \frac{1}{r^2}\sin(\theta)cos(\theta)\partial_\theta + r^{-1}\cos^2(\theta)\partial_r + 2r^{-1}\sin{\theta}\cos{\theta}\partial_r\partial_\theta +
r^{-2}\sin^2(\theta)\partial_\theta^2+r^{-2}\sin(\theta)\cos(\theta)\partial_\theta\bigr)=\\
\partial_r^2 +\frac{1}{r}\partial_r +\frac{1}{r^2}\partial_\theta^2.
\end{multline*}

Title: Re: Change of Variable
Post by: Victor Ivrii on November 05, 2016, 08:54:10 AM

OK. Please fix your screen name!

Title: Re: Change of Variable
Post by: Luyu CEN on November 08, 2016, 08:56:21 PM

Fixed!