User:Egm6341.s10.team3.heejun/Meeting 30 transcript

 '''MTG 30: Tue. 16 Mar 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:58, 11 August 2010 (UTC)


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 * Summary: Error of Trapezoidal
 *   (1)  [[media: Egm6341.s10.mtg27.djvu | Page 27-1]]
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1) $$ \displaystyle \bar{d_{2i}} = \frac{P_{2i}(+1)}{2^{2i}} $$  $$ \displaystyle P_{1}(t) = -t $$  $$ \displaystyle (P_{2i},\,\,  P_{2i+1}) $$ obtained by integration by parts  $$ \displaystyle P_{2i+1}(t) = \int P_{2i} (t) dt $$  $$ \displaystyle P_{2i+1} $$ odd function such that $$ \displaystyle P_{2i+1}(0) = 0,\,\,  P_{2i+1}(\pm1)=0 $$  $$ \displaystyle P_{2i} $$ even function.
 * Composite of $$ \displaystyle \bar{d_{2i}} $$
 * Composite of $$ \displaystyle \bar{d_{2i}} $$
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$$ \displaystyle x\,\, coth\,\, x $$
 * 2) $$ \displaystyle \bar{d_{2i}} = - \frac{B_{2i}}{(2i)!}\,\,\,\, $$ Taylor series of
 * 2) $$ \displaystyle \bar{d_{2i}} = - \frac{B_{2i}}{(2i)!}\,\,\,\, $$ Taylor series of
 *  [[media: Egm6341.s10.mtg28.djvu | Page 28-2]]
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 * 3) obtain $$ \displaystyle (P_{2i},\,\, P_{2i+1}) $$  using recurrence formula
 *   (6)  [[media: Egm6341.s10.mtg29.djvu | Page 29-2]]
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HW: Use $$ \displaystyle (P_{2},\, P_{3}),\,\, (P_{4},\, P_{5}), \,\, (P_{6},\, P_{7}) $$ to understand the Kessler's code
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Arc Length


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$$ \displaystyle \bar{OB} = r (\theta +d \theta) $$ $$ \displaystyle \bar{AB} = dl $$ infinite arc length
 * $$ \displaystyle \bar{OA} = r(\theta) $$
 * $$ \displaystyle \bar{OA} = r(\theta) $$
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Goal: Find $$ \displaystyle dl $$ in terms of $$ \displaystyle (r,\, \theta),\,d \theta. $$
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 * $$ \displaystyle \bar{AB}^{2} = dl^{2} = \bar{AA'}^{2} + \bar{A'B}^{2} $$
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$$ \displaystyle \bar{A'B} = dr \cong r(\theta + d \theta) - r(\theta) \cong \frac{dr(\theta)}{d \theta}d \theta + hot $$
 * $$ \displaystyle \bar{AA'} = r(\theta) d \theta $$
 * $$ \displaystyle \bar{AA'} = r(\theta) d \theta $$
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 * $$ \displaystyle dl^{2} = r^{2}d \theta^{2} + dr^{2} = d \theta^{2} (r^{2} + (\frac{dr}{d \theta})^{2}) $$
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 * $$ \displaystyle dl = d \theta [r^{2}+(\frac{dr}{d \theta})^{2}]^{\frac{1}{2}} $$
 *   (1)
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 * $$ \displaystyle Arclength (PQ) = \int^{\theta_{Q}}_{\theta_{P}} d \theta [r^{2}+(\frac{dr}{d \theta})^{2}]^{\frac{1}{2}} $$
 *   (2)
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 * $$ \displaystyle Ellipse:\,\, r(\theta) = \frac{a(1-e^{2})}{1-ecos \theta} $$
 *   (3)
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a=1 [[media: Egm6341.s10.mtg25.djvu | Page 25-2]]


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 * $$ \displaystyle Circumference:\,\, \int^{\theta=2\pi}_{\theta=0} dl = C $$
 *   (4)
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[[media: Egm6341.s10.mtg30.djvu | Page 30-4]] 


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 * $$ \displaystyle C = 4aE(e^{2}) $$
 *   (5)
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Elliptic integral of the second kind


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 * $$ \displaystyle E(e^{2}) = \int^{\frac{\pi}{2}}_{0} [1-e^{2}sin \theta]^{\frac{1}{2}} d \theta $$
 *   (6)
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HW: Composite C using 1)  (4)  [[media: Egm6341.s10.mtg30.djvu | Page 30-3]]   2)   (5) + (6)  [[media: Egm6341.s10.mtg30.djvu | Page 30-4]] Using 3 most efficient codes (methods) a) Composite Trapezoidal b) Romberg c) Clencurt
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