User:Egm6321.f09.Team1.sallstrom/HW5

= Problem 1: Laplace' Operator = From lecture note slide 29-4.

Given
Laplace' operator acting on $$\displaystyle \psi$$ can be written
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\Delta \psi = \frac{1}{h_1 h_2 h_3} \sum_{i=1}^3 \frac{\partial}{\partial \xi_i} \left[ \frac{h_1 h_2 h_3}{h_i^2} \frac{\partial \psi}{\partial \xi_i} \right] $$ $$ Using
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }
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h_1 = 1, \quad h_2 = r \cos \theta, \quad h_3 = r $$ $$ and
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
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 * }
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\xi_1 = r, \quad \xi_2 = \varphi, \quad \xi_3 = \theta $$ $$ it can be shown that the first term in Eq. 1 is
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
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 * }
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\frac{1}{r^2 \cos \theta} \frac{\partial}{\partial r} \left[ \frac{r^2 \cos \theta}{1} \frac{\partial \varphi}{\partial r} \right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
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Find
Which are the to remaining terms of Eq. 1?

Solution
The second term is
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\Delta \psi_2 \frac{1}{h_1 h_2 h_3} \frac{\partial}{\partial \xi_2} \left[ \frac{h_1 h_2 h_3}{h_2^2} \frac{\partial \psi}{\partial \xi_2} \right] = \frac{1}{r^2 \cos \theta} \frac{\partial}{\partial \varphi} \left[ \frac{r^2 \cos \theta}{r^2 \cos^2 \theta} \frac{\partial \psi}{\partial \varphi} \right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
 * }
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\Delta \psi_2 = \frac{1}{r^2 \cos \theta} \frac{\partial^2 \psi}{\partial \varphi^2} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 6)
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 * }

The third term is
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\Delta \psi_3 = \frac{1}{h_1 h_2 h_3} \frac{\partial}{\partial \xi_3} \left[ \frac{h_1 h_2 h_3}{h_3^2} \frac{\partial \psi}{\partial \xi_3} \right] = \frac{1}{r^2 \cos \theta} \frac{\partial}{\partial \theta} \left[ \frac{r^2 \cos \theta}{r^2} \frac{\partial \psi}{\partial \theta} \right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 7)
 * }
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\Delta \psi_3 = \frac{1}{r^2 \cos \theta} \left[ \sin \theta \frac{\partial \psi}{\partial \theta} + \cos \theta \frac{\partial^2 \psi}{\partial \theta^2} \right] $$ $$ or
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 8)
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 * }
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$$\displaystyle \Delta \psi_3 = \frac{\tan \theta}{r^2} \frac{\partial \psi}{\partial \theta} + \frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2} $$ $$
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 * $$\displaystyle (Eq. 9)
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= Problem 2: Euler's Equation = From lecture note slide 30-2

Given
The Euler equation is
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r^2 R'' + 2 r R' - k R = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

Find
Using the trial solution


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R = r^\lambda $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
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 * }

show that


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\lambda(\lambda + 1) = k $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
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 * }

Solution
Differentiate Eq. 2 twice
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R' = \lambda r^{\lambda - 1} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
 * }
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R'' = \lambda (\lambda - 1) r^{\lambda - 2} $$ $$ Insert Eqs. 2, 4 and 5 into Eq. 1
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
 * }
 * }


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\lambda (\lambda - 1) r^{\lambda} + 2 \lambda r^{\lambda} - k r^\lambda = 0 $$ $$ Divide by $$\displaystyle r^\lambda$$ and rearrange
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 6)
 * }
 * }
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k = \lambda (\lambda - 1) + 2 \lambda $$ $$ or
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 7)
 * }
 * }
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$$\displaystyle \lambda (\lambda + 1) = k $$ $$
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 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * <p style="text-align:right;">$$\displaystyle (Eq. 8)
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