User:Egm6321.f10.team2.oztekin/hw7.1

R*7.1 - Finding ds and Laplace Operator Equivalent in Spherical Coordinates
From the lecture slide Mtg 39-1

Given
Infinitesimal length in Cartesian coordinates given as in Mtg 38-5 ;


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$$\begin{align} ds=dx_{j}.e_{j} \end{align}$$ (7.1.1)
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$$\begin{align} ds^{2}=ds.ds=\left ( dx_{j}.e_{j} \right )\left ( dx_{i}e_{i} \right )=dx_{i}.dx_{j}(e_{i}e_{j}) \end{align}$$ (7.1.2)
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$$\begin{align} e_{i}e_{j}=\delta _{ij} \end{align}$$ (7.1.3)
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Delta function defined as ;


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$$\begin{align} \delta _{ij}=\left \{ \begin{matrix} 1 & for & i=j\\ 0& for  & i\neq j \end{matrix} \right. \end{align}$$ (7.1.4)
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$$\begin{align} ds^{2}=dx_{i}.dx_{i}=\sum_{i=1}^{3}\left ( dx_{i} \right )^{2} \end{align}$$ (7.1.5)
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$$\begin{align} &x_{1}=x=rcos(\theta )cos(\phi )=\xi _{1}cos(\xi _{2})cos(\xi _{3}) \\ &x_{2}=y=rcos(\theta )sin(\phi )=\xi _{1}cos(\xi _{2})sin(\xi _{3}) \\ &x_{3}=z=rsin(\theta )=\xi _{1}sin(\xi _{2}) \\

\end{align}$$ (7.1.6)
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$$\begin{align}

\Delta u =\left [ \frac{1}{h_{1}h_{2}h_{3}} \right ]\sum_{i=1}^{3}\frac{\partial }{\partial \xi _{i}}\left [ \frac{h_{1}h_{2}h_{3}}{h_{i}^{2}} \frac{\partial u }{\partial \xi _{i}}\right ]

\end{align}$$ (7.1.7)
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$$\begin{align} \begin{matrix} \hat{\xi }=(\hat{\xi _{1}},\hat{\xi _{2}},\hat{\xi _{3}}) \end{matrix} \end{align}$$ (7.1.8)
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$$\begin{align} \begin{matrix} h_{1}h_{2}h_{3}=r^{2}cos(\theta ) \end{matrix} \end{align}$$ (7.1.9)
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Find

 * Show that infinitesimal length 'ds' can be written as (5) in meeting 39-1 in spherical coordinates.
 * Derive Laplace operator in spherical coordinates.

Solution

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$$\begin{align} dx_{1}=\left [ \frac{\partial x_{1}}{\partial r} \right ]dr+\left [ \frac{\partial x_{1}}{\partial \theta } \right ]d\theta +\left [ \frac{\partial x_{1}}{\partial \phi } \right ]d\phi

\end{align}$$ (7.1.10)
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$$\begin{align} dx_{2}=\left [ \frac{\partial x_{2}}{\partial r} \right ]dr+\left [ \frac{\partial x_{2}}{\partial \theta } \right ]d\theta +\left [ \frac{\partial x_{2}}{\partial \phi } \right ]d\phi \end{align}$$ (7.1.11)
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$$\begin{align} dx_{3}=\left [ \frac{\partial x_{3}}{\partial r} \right ]dr+\left [ \frac{\partial x_{3}}{\partial \theta } \right ]d\theta +\left [ \frac{\partial x_{3}}{\partial \phi } \right ]d\phi \end{align}$$ (7.1.12)
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$$\begin{align} dx_{1}=\frac{\partial r}{\partial r}cos(\theta )cos(\phi )dr+r\frac{\partial cos(\theta )}{\partial \theta }cos(\phi )d\theta +rcos(\theta )\frac{\partial cos(\phi )}{\partial \phi }d\phi

\end{align}$$ (7.1.13)
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$$\begin{align} dx_{2}=\frac{\partial r}{\partial r}cos(\theta )sin(\phi )dr+r\frac{\partial cos(\theta )}{\partial \theta }sin(\phi )d\theta +rcos(\theta )\frac{\partial sin(\phi )}{\partial \phi }d\phi \end{align}$$ (7.1.14)
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$$\begin{align} dx_{3}=\frac{\partial r}{\partial r}sin\theta dr+r\frac{\partial sin(\theta )}{\partial \theta }d\theta \end{align}$$ (7.1.15)
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$$\begin{align} dx_{1}^{2}=&(cos\theta)^{2} (cos\phi)^{2} dr^{2}+r^{2}(sin\theta)^{2} (cos\phi )^{2}d\theta ^{2}+r^{2}(cos\theta)^{2} (sin\phi)^{2} d\phi ^{2}+2r^{2}sin\phi cos\phi (sin\theta) (cos\theta) d\theta d\phi \\ &-2r(cos\theta) (cos\phi)^{2} (sin\theta) drd\theta -2r(cos\theta) (cos\phi) (sin\phi) drd\phi

\end{align}$$ (7.1.16)
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$$\begin{align} dx_{2}^{2}=&(cos\theta)^{2} (sin\phi)^{2} dr^{2}+r^{2}(sin\theta)^{2} (sin\phi )^{2}d\theta^{2} +r^{2}(cos\theta)^{2} (cos\phi)^{2} d\phi ^{2}+2r(cos\theta )^{2}(sin\phi) (cos\phi) drd\phi \\ &-2r^{2}(sin\theta) (sin\phi) (cos\theta) (cos\phi )d\theta d\phi -2r(cos\theta) (sin\theta) sin^{2}\phi drd\theta \end{align}$$ (7.1.17)
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$$\begin{align} dx_{3}^{2}=(sin\theta)^{2} dr^{2}+r^{2}(cos\theta )^{2}d\theta ^{2}+2r(sin\theta ) (cos\theta) drd\theta \end{align}$$ (7.1.18)
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$$\begin{align} ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2} \end{align}$$ (7.1.19)
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$$\begin{align} ds^{2}=&{\color{Green} (cos\theta)^{2} (cos\phi)^{2} dr^{2}}+{\color{Blue} r^{2}(sin\theta)^{2} (cos\phi)^{2} d\theta ^{2}}+{\color{Cyan} r^{2}(cos\theta)^{2} (sin\phi)^{2} d\phi ^{2}}+{\color{Green} (cos\theta)^{2} (sin\phi)^{2} dr^{2}}+{\color{Blue} r^{2}(sin\theta)^{2} (sin\phi)^{2} d\theta^{2}}\\ & +{\color{Cyan} r^{2}(cos\theta)^{2} (cos\phi)^{2} d\phi ^{2}}+(sin\theta)^{2} dr^{2}+r^{2}(cos\theta)^{2} d\theta ^{2}+{\color{Red} 2r^{2}sin\phi cos\phi sin\theta cos\theta d\theta d\phi} -2rcos\theta (cos\phi)^{2} sin\theta drd\theta \\ & -{\color{Orange} 2r(cos\theta)^{2} cos\phi sin\phi drd\phi}+{\color{Orange} 2r(cos\theta)^{2} sin\phi cos\phi drd\phi }-{\color{Red} 2r^{2}sin\theta sin\phi cos\theta cos\phi d\theta d\phi} -2rcos\theta sin\theta sin^{2}\phi drd\theta \\ &+2rsin\theta cos\theta drd\theta \end{align}$$ (7.1.20)
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If we group things together we will get ;

$$  \displaystyle ds^{2}=1.dr^{2}+r^{2}d\theta ^{2}+r^{2}(cos\theta)^{2} d\phi ^{2} $$


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$$\begin{align} &i=1\\ &\frac{\partial }{\partial \xi _{1}}\left [ \frac{h_{1}h_{2}h_{3}}{h_{1}^{2}} \frac{\partial u }{\partial \xi _{1}}\right ]=\frac{\partial }{\partial r}\left [ \frac{r^{2}cos(\theta )}{(1)^{2}} \frac{\partial u }{\partial r}\right ]

\end{align}$$

(7.1.21)
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$$\begin{align} &i=2\\ &\frac{\partial }{\partial \xi _{2}}\left [ \frac{h_{1}h_{2}h_{3}}{h_{2}^{2}} \frac{\partial u }{\partial \xi _{2}}\right ]=\frac{\partial }{\partial \theta }\left [ \frac{r^{2}cos(\theta )}{(r)^{2}} \frac{\partial u }{\partial \theta }\right ] \end{align}$$ (7.1.22)
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$$\begin{align} &i=3\\ &\frac{\partial }{\partial \xi _{3}}\left [ \frac{h_{1}h_{2}h_{3}}{h_{3}^{2}} \frac{\partial u }{\partial \xi _{3}}\right ]=\frac{\partial }{\partial \phi }\left [ \frac{r^{2}cos(\theta )}{(r)^{2}(cos\theta )^{2}} \frac{\partial u }{\partial \phi }\right ] \end{align}$$ (7.1.23)
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If we substitute in


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$$\begin{align} \Delta u =\frac{1}{r^{2}cos(\theta )}\left [ \frac{\partial }{\partial r}\left [ \frac{r^{2}cos(\theta )}{(1)^{2}} \frac{\partial u }{\partial r}\right ]+\frac{\partial }{\partial \theta }\left [ \frac{r^{2}cos(\theta )}{(r)^{2}} \frac{\partial u }{\partial \theta }\right ]+\frac{\partial }{\partial \phi }\left [ \frac{r^{2}cos(\theta )}{(r)^{2}(cos\theta )^{2}} \frac{\partial u }{\partial \phi }\right ] \right ] \end{align}$$ (7.1.24)
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$$  \displaystyle \Delta u =\frac{1}{r^{2}}\left ( r^{2}\frac{\partial u }{\partial r} \right )+\frac{1}{r^{2}cos\theta }\frac{\partial }{\partial \theta }\left ( cos\theta \frac{\partial u }{\partial \theta } \right )+\frac{1}{r^{2}(cos\theta)^{2} }\frac{\partial ^2 u }{\partial \phi ^2} $$

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