User:EGM6341.s11.TEAM1.WILKS/Mtg33

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_04_13_57_39.djvu | Mtg 33:]] Thu, 4 Nov 10

= Examples: Non-homogeneous L2-ODE-VC (cont'd) =

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HW6.2 continued F09 F10 END HW6.2

HW6.3 Non-homogeneous Legendre Equation $$ n=1 \ $$ [[media: 2010_09_02_13_55_50.djvu | Eq.(1)p.5-4]]: [[media: 2010_09_02_14_58_46.djvu | Eq.(1)p.6-1]]

See K.p.34 for the example. Note that there was a misprint in King 2003 p.34: The excitation function should be


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$$  \displaystyle f(x) = 1 $$     (4)
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and NOT


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$$  \displaystyle f(x) = \frac{1}{1-x^2} $$     (5)
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Solve Eq.(3) using direct method ([[media: 2010_10_26_15_07_13.djvu | Mtg29]] and [[media: 2010_10_28_14_00_19.djvu | Mtg30]] ; See [[media: 2010_10_26_15_07_13.djvu | Eq.(2)p.29-3]])

END HW6.3 [[media: 2010_10_28_14_00_19.djvu | Eq.(2)p.30-2]] $$ u_1(x) = x \equiv P_1(x) \ $$.

= Heat conduction on a sphere =

Transient heat equation:

where $$ \mathbf{k} \ $$ is the conductivity tensor (think matrix) For a homogeneous isotropic material: $$ \mathbf{k} = k \mathbf{I} \ $$, where $$ \mathbf{I} \ $$ is an identity matrix For steady state: $$ \frac{\partial \psi}{\partial t} =0 \ $$ For no heat source: $$ f =0 \ $$ From Eq.(4) $$ \Rightarrow \ k\ {\rm div} ( {\rm grad} \, \psi ) = 0 \ $$ ,

where

$$ {\rm div} ( {\rm grad} \, \psi ) = \nabla \cdot \nabla \psi = \nabla^2 \psi = \Delta \psi \ $$

is the Laplace operator acting on $$\displaystyle \psi$$.

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Repeated index of $$ x_i \ $$ in Eq.(1) is a summation convention $$ m= \ $$ space dimension (1, 2, or 3) Consider an infinitesimal segment $$\displaystyle \mathbf{ds}$$ (vector quantity)



$$ \mathbf{ds} = dx_i \mathbf{e}_i \Rightarrow \ ds^2= \mathbf{ds} \cdot \mathbf{ds} = (dx_i \mathbf{e}_i) \cdot (dx_j \mathbf{e}_j) = dx_i \, dx_j (\mathbf{e}_i \cdot \mathbf{e}_j) \ $$ where $$ \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij} \ $$, which is the Kronecker Delta defined as follows

$$ \Rightarrow \ ds^2= dx_i dx_i = \sum_{i=1}^3 (dx_i)^2 \ $$

Spherical coordinates
Orthogonal curvilinear coordinates: Spherical coordinates $$ ( \xi_1, \xi_2, \xi_3 ) \ $$ (General curvilinear coordinates: $$ ( \xi_1, \xi_2, \xi_3 ) \ $$)

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Astronomy convention: $$ ( r, \theta, \varphi ) \ $$ Mathematical-physics convention: $$ ( r, \bar \theta, \varphi ) \ $$ where $$ \bar \theta := \frac{\pi}{2} - \theta \ $$ From F09: $$ \xi_1 =r \ $$ $$ \xi_2 = \varphi\ \ $$ $$ \xi_3 = \theta\ \ $$