User:Egm6322.s12.team2.steele.m2/Mtg41

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

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Plan: $$\Rightarrow$$Laplacian in elliptic coordinates $$\Rightarrow$$Laplacian in parabolic coordinates $$\Rightarrow$$Heat conduction on a sphere (cont’d) Axisymmetric problem p.39-4 Solution of 1st separated eq. Solution of 2nd separated eq. 1st homogeneous solutions of Legendre eq. Solution of heat equation on a sphere

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R*7.4: Laplacian in elliptic coordinates See graph showing tangent vector, ellipse, and hyperbola http://en.wikipedia.org/wiki/File:Elliptical_coordinates_grid.svg Read the Wikipedia article: Elliptic coordinates $$(\mu, \nu)$$ http://en.wikipedia.org/wiki/Elliptic_coordinates Verify the Laplacian in elliptic coordinates given by

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R7.5: Laplacian in parabolic coordinates Parabola			Tangent vector See graph on this page. http://en.wikipedia.org/wiki/File:Parabolic_coords.svg Read the Wikipedia article: Parabolic coordinates $$(\mu,nu)$$ http://en.wikipedia.org/wiki/Parabolic_coordinates verify the Laplacian in parabolic coordinats given by

Ellptic coordinates and parabolic coordinates are two examples of orthogonal curvilinear coordinates.

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=Heat Conduction on a Sphere (cont’d)= Note: Axisymmetric problem p.39-4 Q: what if axis of symmetry is arbitrarily oriented (i.e. no longer the South-to-North axis)? A: Change the coordinate system such that the new South-to-North axis coincides with new axis of symmetry => Same equation Solution of 1st separated equations [[media:Pea1.f11.mtg39.djvu|Eqn(1) p.39-6]] Note [[media:Pea1.f11.mtg40.djvu|Eqn(5) p.40-2]] and [[media:Pea1.f11.mtg40.djvu|Eqn(2) p.40-4]]:

HW*: Solve (1) to obtain (2). (1)	P. 40-2: Superposition of solutions

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Now the 2nd separated eq. (5) p.40-3 = (1) p.40-4 [[media:Pea1.f11.mtg40.djvu|Eqn(5) p.40-3 & Eqn(1) p.40-4]], which becomes the Legendre eq. [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] if the condition (2) p.40-4 holds. See Note p.40-1 on Legendre funtions as basis functions. $${P_n(x)}$$ 1st homogeneous solutions of Legendre eq.

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[n/2] := integer part of $$\frac{n}{2}$$ R*7.6: Verify that [[media:Pea1.f11.mtg41.djvu|Eqns(1)-(4) p.41-4]]are homogeneous solutions of the Legendre eq. (1) p.7-1. Show that (5) p.41-4 can also be written as

Verify that (1)-(4) p.41-4 can be obtained from (5) p.41-4 or (2). $${Q_n(x)}$$ 2nd homogeneous solutions of Legendre eq. See Example p.37-3.

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Solution of 2nd separated eq. (5) p.40-3 = (1) p.40-4, i.e., homogeneous Legendre equation: For each value of n:

$$\mu := \sin \theta$$			(7) p.40-2 Solution of heat equation on a sphere: (1)	P. 30-5 and superposition of solutions in (1) [[media:Pea1.f11.mtg30.djvu|Eqn(1) p.30-5]]: $$R_n({r})$$ given by (3) p.41-3 $$\Theta_{(\theta)}$$ given by (1) R*7.7: Try to obtain the separated equations for the Laplace in: Elliptic coordinates F10 Parabolic coordinates F11