Mtg 33: Thu, 4 Nov10

HW6.2 continued

F09

F10

${\displaystyle \displaystyle {\begin{aligned}f(x)=\sin x\end{aligned}}}$

(2)

END HW6.2

HW6.3 Non-homogeneous Legendre Equation ${\displaystyle n=1\ }$

Eq.(1)p.5-4:

${\displaystyle \displaystyle {\begin{aligned}(1-x^{2})y''-2xy'+2y={\frac {1}{1-x^{2}}}\end{aligned}}}$

(3)

Eq.(1)p.6-1 (See K.p.84 Example)

Solve Eq.(3) using direct method ( Mtg29 and Mtg30 ; See Eq.(2)p.29-3) END HW6.3

Eq.(2)p.30-2 ${\displaystyle E_{1}(x)=u_{1}(x)=x\ }$

Heat conduction on a sphere; Heat equation is transient

${\displaystyle \displaystyle {\begin{aligned}div(\mathbf {k} grad\psi \ )+f=\rho \ c{\frac {\partial \psi \ }{\partial t}}\end{aligned}}}$

(4)

Where ${\displaystyle \mathbf {k} \ }$ is the conductivity tensor (think matrix)

For a homogeneous isotropic material: ${\displaystyle \mathbf {k} =k\mathbf {I} \ }$ , where ${\displaystyle \mathbf {I} \ }$ is an identity matrix

For steady state: ${\displaystyle {\frac {\partial \psi \ }{\partial t}}=0\ }$

For no heat source: ${\displaystyle f=0\ }$

From Eq.(4) ${\displaystyle \Rightarrow \ k\ div\ grad\ \psi \ =0\ }$ , where ${\displaystyle div\ grad\ \psi \ =\nabla \ \cdot \nabla \ \psi \ =\nabla \ ^{2}\psi \ =\Delta \ \psi \ }$

Repeated index of ${\displaystyle x_{i}\ }$ in Eq.(1) is a summation convention

${\displaystyle m=\ }$ space dimension (1,2 or 3)

Consider an infinitesimal segment ds

${\displaystyle dx_{i}\mathbf {e} _{i}\Rightarrow \ ds^{2}=d\mathbf {s} \cdot d\mathbf {s} =(dx_{i}\mathbf {e} _{i})-(dx_{j}\mathbf {e} _{j})=dx_{i}dx_{j}(\mathbf {e} _{i}\cdot \mathbf {e} _{j})\ }$

Where ${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}\ }$ Kronecker Delta

${\displaystyle \Rightarrow \ ds^{2}=dx_{i}dx_{i}=\sum _{i=1}^{3}(dx_{i})^{2}\ }$

Orthogonal curviture coordinates: Spherical coordinates ${\displaystyle {\delta _{1},\delta _{2},\delta _{3}}\ }$

(General curviture coordinates:{${\displaystyle \delta _{1},\delta _{2},\delta _{3}\ }$}

Astronomy conversion: ${\displaystyle (r,\theta \ ,\varphi \ )\ }$

Math/physics conversion: ${\displaystyle (r,{\overline {\theta \ }}\ ,\varphi \ )\ }$

Where ${\displaystyle {\overline {\theta \ }}:={\frac {\pi \ }{2}}-\theta \ \ }$

From F09:

${\displaystyle \xi _{1}=r\ }$

${\displaystyle \xi _{2}=\varphi \ \ }$

${\displaystyle \xi _{3}=\theta \ \ }$