User:EGM6321.F10.TEAM1.WILKS/EGM6321.F10.TEAM1.WILKS/Mtg31

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 31: Tues, 3Nov09

[[media: Egm6321.f09.mtg31.djvu | Page 31-1]]
HW: Circular cylindrical coordinates



$$ x=r \cos \theta\ = \xi\ _1 \cos \xi\ _2 \ $$

$$ y=r \sin \theta\ = \xi\ _1 \sin \xi\ _2 \ $$

$$ z= \xi\ _3 \ $$

1) Find $$ \left \{ dx_i \right \} = \left \{ dx_1, dx_2, dx_3 \right \} \ $$ in terms of $$ \left \{ \xi\ _j \right \} = \left \{ \xi\ _1, \xi\ _2, \xi\ _3 \right \}  \ $$ and $$ \left \{ d \xi\ _k \right \}  \ $$

2) Find $$ ds^2 = \sum_{i} (dx_i)^2 = \sum_{k} (h_k)^2(d \xi\ _k)^2 \ $$. Identify $$ \left \{ h_i \right \} \ $$ in terms of $$ \left \{ \xi\ _j \right \}  \ $$

3) Find Laplacian in this curviture coordinate

Note: Prelude to Bessel functions; Bessel Differential Equation

HW: Spherical coordinates. Repeat 1), 2) and 3) above steps. See [[media: Egm6321.f09.mtg29.djvu |  P.29-3 ]], use astronautical convention

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Conventions:

Astronomy: $$ \left ( r, \varphi, \theta\ \right ) \ $$ [[media: Egm6321.f09.mtg29.djvu  |  P.29-3 ]]

Math/Physics: $$ \left ( r, \varphi, \bar \theta\ \right ) \ $$, where $$ \bar \theta\ = \frac{ \pi\ }{2} - \theta\ \ $$, [[media: http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false  | K p39 ]]

HW: Find Laplacian in Math/Physics convention. $$ \left ( r, \varphi, \bar \theta\ \right ) \ $$

Continued from [[media: Egm6321.f09.mtg30.djvu |  P.30-4]] : Heat problem in sphere

From [[media: Egm6321.f09.mtg30.djvu | Eq.(2) P.30-4]] and [[media: Egm6321.f09.mtg30.djvu  | Eq.(5) P.30-2]] :

Where:

and

HW: Solve Eq(1) (quadratic equation)

[[media: Egm6321.f09.mtg30.djvu | Eq.(3) P.30-2]] : $$ R(r)=A_nr^n+B_nr^{-(n+1)}  \ $$

One of two homogeneous solutions of Legendre differential equations, [[media: Egm6321.f09.mtg30.djvu | Eq.(1) P.30-4]] is the Legendre ppolynomial $$ P_n( \mu\ ) \ $$. How...given later

[[media: Egm6321.f09.mtg31.djvu | Page 31-3]]
HW: show that Eq(6) = Eq(7)

$$ \left [ \frac{n}{2} \right ] = \ $$ integer part of  $$ \frac{n}{2} \ $$

HW: Verify that Eq(1) thru Eq(5) can be written as Eq(6) or Eq(7)