User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg38

Mtg 38: Tue, 17 Nov 09

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Historical development of $$ \color{blue} \left \{ \mathbf{P}_{n} \right \} \ $$

Legendre: Attraction of Spheres


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Newtonian potential:


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$$ V= \frac{1}{r_{PQ}} \ $$

(1)
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recall force = gradient potential $$ V \ $$

Find $$ r_{PQ} \ $$ in terms of spherical coordinate of point $$ (P,Q) \ $$


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$$ V= \frac{1}{r_{PQ}}= \frac{1}{r_{Q}}( \underbrace{1-2\mu\rho+\rho^{2}}_{ \color{red} Generating \ function \ of \ \left \{ P_{n} \right \}})^{ -\frac{1}{2}} \ $$

$$ \mu:= \cos \gamma, \ \gamma:= \angle(OP,OQ), \ \rho:= \frac{r_{P}}{r_{Q}} \ $$

(2)
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[[media: Egm6321.f09.mtg32.djvu| page38-2]]


 * PEA1.F09.Mtg38.pg2.fig1.svg $$ \color{blue} P=(x_{P},y_{P},z_{P})=(r_{P},\varphi_{P},\theta_{P}), \ Q=(x_{Q},y_{Q},z_{Q})=(r_{Q},\varphi_{Q},\theta_{Q}) \ $$


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$$ (r_{PQ})^{2}= \left \| \vec{PQ} \right \|^{2}= \sum_{i=1}^{3}(x_{Q}^{i}-x_{P}^{i})^{2} \ $$

(1)
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$$ x_{P}^{1}=x_{P}, \ x_{P}^{2}=y_{P}, \ x_{P}^{3}=z_{P} \ $$


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$$ x_{P}^{1}=r_{P} \cos \theta_{P} \cos \varphi_{P} \ \ $$    Astronomical conversion $$ x_{P}^{2}=r_{P} \cos \theta_{P} \sin \varphi_{P} \ $$ $$ x_{P}^{3}=r_{P} \sin \theta_{P} \ $$

(2)
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Similarly for point $$ Q \ $$

(3)
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HW:

Use (2) and (3) in (1) to find $$ (r_{PQ})^{2} \ $$


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[[media: Egm6321.f09.mtg38.djvu| page38-3]]


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$$ (r_{PQ})^{2}=(r_{P})^{2}+(r_{Q})^{2}-2r_{P}r_{Q} \cos \gamma \ $$

(1)
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$$ \cos \gamma = \cos \theta_{Q} \cos \theta_{P} \cos( \varphi_{Q}- \varphi_{P})+ \sin \theta_{Q} \theta_{P} \ $$

(2)
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$$ (r_{PQ})^{2}=(r_{Q})^{2} \left [ \underbrace{( \frac{r_{P}}{r_{Q}})^{2}}_{ \color{blue} \rho^{2}}+1-2( \frac{r_{P}}{r_{Q}}) \underbrace{ \cos \gamma}_{ \color{blue} \mu} \right ] \ $$ obtain [[media: Egm6321.f09.mtg38.djvu| (2)page38-1]]

(3)
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$$ V= \frac{1}{r_{PQ}}= \frac{1}{r_{Q}} \sum_{n=0}^{ \infty} P_{n}( \mu) \rho^{n} \ $$

(4)
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$$ ( \underbrace{1-2 \mu \rho+\rho^{2}}_{ \color{blue} := -x})^{ -\frac{1}{2}}= \sum_{n=0}^{ \infty} P_{n}( \mu) \rho^{n} \ $$

(5)
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Binomial Theorem: $$ (1-x)^{- \frac{1}{2}}= \sum_{i=0}^{ \infty} \alpha_{i} x^{i} \ $$

(6)
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$$ \alpha_{i}= \frac{1 \cdot 3 \cdot \cdots (2i-1)}{2 \cdot 4 \cdot \cdots (2i)} \ $$

(7)
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[[media: Egm6321.f09.mtg38.djvu| page38-4]]


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$$ (1-x^{2})^{-1}=1+x^{2}+ \cdots \ $$ K. p.35

(1)
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Binomial Theorem (more generation): n integration


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$$ (x+y)^{n}= \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^{k} \ $$

(2)
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$$ \binom{n}{k}= \frac{n!}{k!(n-k)!}= \frac{n(n-1) \cdots (n-k+1)}{k!} \ $$

(3)
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Generalize (2) to case $$ n=r= \ $$real number


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$$ (x+y)^{r}= \sum_{k=0}^{ \infty} \binom{r}{k} x^{r-k} y^{k} \ $$

(4)
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$$ \binom{r}{k}= \frac{r(r-1) \cdots (r-k+1)}{k!} \ $$

(5)
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HW:

Use (4) and (5) to obtain (6) and (7) [[media: Egm6321.f09.mtg38.djvu| page38-3]].