User:Egm6322.s12.team2.steele.m2/Mtg39

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Plan: $$\Rightarrow$$Heat conduction on a sphere (cont’d) Cartesian coord. In terms of spherical coord. Infinitesimal length in spherical coord. Laplace operator in general curvilinear coord. Laplace operator in spherical coordinates Axisymmetric problems Laplace equation: Homog. Heat eq. Separation of variables Separated equations

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=Heat Conduction on a Sphere (cont’d)= Cartesian coord. In terms of spherical coord.:

Using [[media:Pea1.f11.mtg4.djvu|Eqn(3) p.4-3]] and [[media:Pea1.f11.mtg4.djvu|Eqn(1) p.4-4]], (1)-(3) can be written in compact form as

R*7.1: Infinitesimal length in spherical coord. Show that the infinitesimal length ds in [[media:Pea1.f11.mtg38.djvu|Eqn(2) p.38-6]] can be written in spherical coord. As follows

Note $$(h_1)^2 \, (d\xi_1)^2 \, (h_2)^2 \, (d\xi_2)^2 \, (h_3)^2 \, (d\xi_3)^2$$.

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$$h_i$$ = magnitude of the tangent vector $$\mathbf g_i$$ to the coordinate line $$\xi_i$$ See graph for $$g_1 \, g_2 \, g_3 \, \xi_1 \, \xi_2 \, \xi_3$$ in general curvilinear coordinates.

Laplace Operator in General Curvilinear Coordinates=



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=Laplace Operator in Spherical Coordinates=

Note i=1:

Note i=2: R*7.1 p.39-1 Note i=3: R*7.1 p.39-1 Thus, (2) p.39-2:

Note reference to R*7.1 p. 39-1

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For axisymmetric problems, there is no dependence on the longitude $$\varphi$$. Thus,

Cf. King 2003 p.45 (in which the math/physics convention is used, i.e., $$\bar \theta$$ instead of $$\theta$$ ; see [[media:Pea1.f11.mtg38.djvu|Eqn(5) p.38-6]] . Laplace Equation: Homogeneous heat equation

See unit sphere image on this page.

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=Separation of Variables= (See beam vibration, Euler equation L4-ODE-CC [[media:Pea1.f11.mtg30.djvu|p.30-1]]

(1)	And (1) p.39-4:

Cancel $$\frac{1}{r^2}$$ and divide by $$R\Theta$$ :

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Separated equations:

Note (1) is an Euler (homogeneous) L2-ODE-VC, see [[media:Pea1.f11.mtg30.djvu|Eqn(1) p.30-4]]. Note (2) will be shown to lead to the Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] (with a paradox).