User:EGM6341.s11.team1.Chiu/S10 Mtg42

Mtg 42: Tue, 13 Apr 10

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HW:


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show: $$ \overline{CD}= \overline{AB}+hot \ $$     for $$ d \gamma \ $$     small


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Parameterization of ellipse:


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Eliminate t     $$ \Rightarrow \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 \ $$      HW


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Arc length:     $$ dl= \left [ dx^{2}+dy^{2} \right ]^{ \frac{1}{2}} \ $$      HW


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Eccentricity    $$ e= \left [ 1- \frac{b^{2}}{a^{2}} \right ]^{ \frac{1}{2}} \ $$


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$$ \Rightarrow c = \int dl = a \int^{2 \pi}_{t=0} \left [ 1-e^{2}cos^{2}t \right ]^{ \frac{1}{2}}dt \ $$     HW


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$$ = 4a \underbrace{ \int^{ \frac{ \pi}{2}}_{ \alpha=0} \left [ 1-e^{2}sin^{2} \alpha \right ]^{ \frac{1}{2}}d \alpha}_{ \color{blue} elliptic \ integration \ 2nd \ kind} \ $$     HW


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End Parameterization of ellipse

Clenshaw-Curtis:

*Periodicity (trapezoidal error)

*Runge phenomenon

Change of variables: $$ x= \underset{ \color{blue} \underset{chebyshev}{ \uparrow}}{cos \theta} \ $$


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$$ I= \int^{+1}_{-1}f(x)dx= \int^{ \pi}_{0}f(cos \theta) \underbrace{sin \theta d \theta}_{ \color{blue} -dx} \ $$     HW


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Integrate simpler if cosine series of     $$ f(cos \theta) \ $$ known.

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$$ f(cos \theta)= \underbrace{ \frac{a_{0}}{2}+ \sum^{ \infty}_{k=1}a_{k}cos(k \theta)}_{\color{blue} cosine \ series} \ $$


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$$ a_{k}= \frac{2}{ \pi} \int^{ \pi}_{0}f(cos \theta)cos(k \theta)d \theta \ $$     HW


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$$ I=a_{0}+ \sum^{ \infty}_{k { \color{blue}\to j}=1} \frac{2a_{2j}}{1-(2j^{2})} \ $$     HW


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$$ a_{k} \ $$     evaluated number using trapezoidal rule which corresponds to the Discrete Cosine Transfer (DCT). (which can be computed efficiently using FFT: Fast Fourier Transfer)