User:Egm6321.f09.Team06.sada/HW7

Problem #5:
From Lecture p.37-3

Show that the following is true, by performing integration by parts:

$$ \begin{align} \alpha = -\int_{-1}^{+1} \left(1-x^{2}\right)L_{n}'L_{m}'\text{d}x \end{align} $$

Solution
Let,


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u = L_m \Rightarrow du = L'_m dx, $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
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and,


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dv = \left[ (1-x^2) L'_n \right]' dx. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
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 * }


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\Rightarrow \int dv = \int\left[ (1-x^2) L'_n \right]' dx \Rightarrow v = \left[ (1-x^2) L'_n \right]. $$
 * $$\displaystyle
 * $$\displaystyle
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We know,


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\int_{a}^{b} udv = \left[uv \right]_{a}^{b} - \int_{a}^{b} vdu , $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
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 * }

and so, with $$\displaystyle a = -1 $$ and $$\displaystyle b = +1 $$


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\alpha = \int_{-1}^1 L_m \left[ (1-x^2) L'_n \right]' dx = \left[ L_m(1-x^2) L'_n\right]_{-1}^{+1} - \int_{-1}^{+1} (1-x^2) L'_n L'_m dx , $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
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For values of $$\displaystyle x = -1 $$ and $$\displaystyle x = +1 $$,


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(1-x^2) = 0. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
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Substituting Equation (5) in first term of Equation (4),


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\alpha = {\left[ L_m(1-x^2) L'_n\right]_{-1}^{+1}} - \int_{-1}^{+1} (1-x^2) L'_n L'_m dx. $$
 * $$\displaystyle
 * $$\displaystyle
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The first term becomes zero,So


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$$\displaystyle \alpha = - \int_{-1}^{+1} (1-x^2) L'_n L'_m dx $$
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Problem #6:
From Lecture p.38-2

$$ \begin{align} \left(r_{pq}\right)^{2} = ||\vec{PQ}||^{2} = \sum_{i=1}^{3}\left(x_{qi}-x_{pi}\right)^{2} \end{align} $$

Points $$P$$ and $$Q$$ in spherical coordinates,

$$ \begin{align} P &= \left(r_{p},\phi_{p},\theta_{p}\right) \\ Q &= \left(r_{q},\phi_{q},\theta_{q}\right) \end{align} $$

The cartesian coordinates of these points be given as:

$$ \begin{align} P &= \left(x_{p1}, x_{p2}, x_{p3}\right) \\ Q &= \left(x_{q1}, x_{q2}, x_{q3}\right) \end{align} $$

The relationship between these two coordinate systems is given as

$$ \begin{align} x_{p1} &= r_{p}\cos\left(\theta_{p}\right)\cos\left(\phi_{p}\right) \\ x_{p2} &= r_{p}\cos\left(\theta_{p}\right)\sin\left(\phi_{p}\right) \\ x_{p3} &= r_{p}\sin\left(\theta_{p}\right) \\ \\ x_{q1} &= r_{q}\cos\left(\theta_{q}\right)\cos\left(\phi_{q}\right) \\ x_{q2} &= r_{q}\cos\left(\theta_{q}\right)\sin\left(\phi_{q}\right) \\ x_{q3} &= r_{q}\sin\left(\theta_{q}\right) \end{align} $$

Convert the following equation for $$\left(r_{pq}\right)^{2}$$ to spherical coordinates, by substituting in the relationships above.

Solution
Equation (1) can be written as,


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(r_{PQ})^2 = \left(x_Q - x_P\right)^2 + \left(y_Q - y_P\right)^2 + \left(z_Q - z_P\right)^2. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 6)
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 * }

Substituting Equations (2) and (3) in (6),


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\begin{align} \Rightarrow (r_{PQ})^2 = & \left[r_Q \cos(\theta_Q) \cos(\psi_Q) - r_P \cos(\theta_P) \cos(\psi_P)\right]^2 + \\ & \left[r_Q \cos(\theta_Q) \sin(\psi_Q) - r_P \cos(\theta_P) \sin(\psi_P)\right]^2 + \\ & \left[r_Q \sin(\theta_Q) - r_P \sin(\theta_P)\right]^2 \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
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 * }


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\begin{align} \Rightarrow (r_{PQ})^2 = & \left[r_Q^2 \cos^2(\theta_Q) \cos^2(\psi_Q) + r_P^2 \cos^2(\theta_P) \cos^2(\psi_P) - 2 r_P r_Q \cos(\theta_Q) \cos(\psi_Q) \cos(\theta_P) \cos(\psi_P) \right] + \\ & \left[r_Q^2 \cos^2(\theta_Q) \sin^2(\psi_Q) + r_P^2 \cos^2(\theta_P) \sin^2(\psi_P) - 2 r_P r_Q \cos(\theta_Q) \sin(\psi_Q) \cos(\theta_P) \sin(\psi_P)\right] + \\ & \left[r_Q^2 \sin^2(\theta_Q) + r_P^2 \sin^2(\theta_P) - 2 r_P r_Q \sin(\theta_Q) \sin(\theta_P)\right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle


 * }
 * }


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\begin{align} \Rightarrow (r_{PQ})^2 = &\ r_Q^2 \cos^2(\theta_Q) {\left[ \cos^2(\psi_Q) + \sin^2(\psi_Q) \right]} + r_Q^2 \sin^2(\theta_Q) -  2 r_P r_Q \cos(\theta_Q) \cos(\theta_P) \left [\cos(\psi_Q) \cos(\psi_P) \right] + \\ &\ r_P^2 \cos^2(\theta_P) {\left[ \cos^2(\psi_P) + \sin^2(\psi_P) \right]} + r_P^2 \sin^2(\theta_P) - 2 r_P r_Q \cos(\theta_Q)\cos(\theta_P) \left [\sin(\psi_Q) \sin(\psi_P)\right] - 2 r_P r_Q \sin(\theta_Q) \sin(\theta_P) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
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\begin{align} \Rightarrow (r_{PQ})^2 = &\ r_Q^2 {\left[\cos^2(\theta_Q) + \sin^2(\theta_Q) \right]} -  2 r_P r_Q \cos(\theta_Q) \cos(\theta_P) \underbrace{\left [\cos(\psi_Q) \cos(\psi_P) + \sin(\psi_Q) \sin(\psi_P)\right]}_{=\ \cos(\psi_Q-\psi_P)} + \\ &\ r_P^2 {\left[ \cos^2(\theta_P) + \sin^2(\theta_P) \right]} - 2 r_P r_Q \sin(\theta_Q) \sin(\theta_P)
 * $$\displaystyle
 * $$\displaystyle

\end{align} $$
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 * }


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\begin{align} \Rightarrow (r_{PQ})^2 = &\ r_Q^2 + r_P^2 -  2 r_P r_Q \left[\underbrace{\cos(\theta_Q) \cos(\theta_P) {\cos(\psi_Q-\psi_P)} + \sin(\theta_Q)\sin(\theta_P)}_{=\ \cos\gamma} \right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
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And so,


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$$\displaystyle (r_{PQ})^2 = (r_P)^2 + (r_Q)^2 - 2 (r_P) (r_Q) \cos\gamma $$
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