User:EGM6321.F10.TEAM1.WILKS/Mtg33

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 33: Thurs, 5Nov09

[[media: Egm6321.f09.mtg33.djvu | Page 33-1]]
Where: $$ \sin \theta\ = \mu\ \ $$ and $$ d \theta\ \ $$ becomes $$ d \mu\ \ $$

Similarly for $$ \left \langle P_n, P_m \right \rangle $$

Orthogonality of Legendre polynomial

Where $$ \delta_{mn} = \ $$ kronecker delta

[[media: Egm6321.f09.mtg33.djvu | Page 33-2]]
$$ \Rightarrow \ \underline{ \Gamma\ } (F) \ $$ is diagonal with diagonal coefficient:

F is complete, i.e. any continuous function, f, can be expressed as an infinite series of function in F:

Eq(4) is an equality due to the completeness of F

p29-5: $$ f( \theta\ )= T_0 \cos ^4 \theta\ = T_0(1- \mu\ ^2)^2\sum_{u=0}^\infty A_nP_n( \mu\ ) \ $$

Where $$ \mu\ = \sin |theta\ \ $$

Where n=0,1,2...n

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HW $$ f= \sum_{i} g_i \ $$

Show that if $$ \left \{ g_i \right \} \ $$ is odd, then f is odd

Show that if $$ \left \{ g_i \right \} \ $$ is even, then f is even END HW

HW Show that $$ P_{2k} \ $$ is even for k=0,1,2... and $$ P_{2k+1} \ $$ is odd END HW

[[media: Egm6321.f09.mtg33.djvu | Eq.(5) P.33-2]] $$ A_n=\frac{ \left \langle f, P_n \right \rangle }{ \left \langle P_n, P_n \right \rangle } \ $$, f even

$$ \Rightarrow \ A_n=0 \ $$ for  $$  n=2k+1 \ $$, since $$  P_{2k+1}(x) \ $$ is odd

$$ A_1=A_3=A_5=...=0 \ $$

It turns out that $$ A_n=0 \ $$ for all $$ n \ge 5 \ $$ due to linear independance of $$ F= \left \{ P_n \right \} \ $$ and the orthogonality of $$ F \ $$

[[media: Egm6321.f09.mtg33.djvu | Page 33-4]]
Linear independance of $$ F \ $$

$$ P_n(x) \ $$ is a polynomial of order n

$$ P_n \in \Rho\ _n \ $$ set of all polynomials of degree (order)  $$ \le \ n \ $$

HW $$ q \in \Rho\ _4 \ $$

$$ q(x) = \sum_{i=0}^4 c_ix^i \ $$

Given $$ c_0=3, c_1=10, c_2=15, c_3=-1, c_4=5 \ $$

Find $$ \left \{ a_i \right \} \ $$ such that $$ q(x) = \sum_{i=0}^4 a_iP_i \ $$

Plot $$ q = \sum_{i} c_ix^i = \sum_{i} a_iP_i \ $$

Where $$ \sum_{i} c_ix^i = \ $$ figure 1 and $$ \sum_{i} a_iP_i= \ $$ figure 2 END HW

Othogonality of $$ F= \left \{ P_k \right \} \ $$ [[media: Egm6321.f09.mtg33.djvu | Eq.(3) P.33-1]]