User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg36

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Mtg 36: Thu, 12 Nov 09

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Let i.e. is a polynomial of degree

Example

(1) page33-4

End Example

i.e., we can integrate exactly any polynomial of degree using only integral points (almost half).

Trapezoidal Rule:

PEA1.F09.Mtg36.pg1.fig1.svg

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number of panels (trapezoidal)

Trapezoidal Rule can only integrate exact a straight line. Otherwise, as even for a simple polynomial of degree 3. (not even degree 2)

End Trapezoidal Rule

Question 1: Origin (2) of Legendre polynomial page31-3 and Legendre equation (1) page14-2.

End Question 1

Question 2: Why solving Laplace equation (heat, fluid,...) in a sphere gave rise to Legendre equation (2)?

End Question 2

Answer 1: Legendre's idea: Expand Newtonian potential into power series in his study of attraction of spheres Legendre polynomial.

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Legendre found the differential equation that admits Legendre polynomial as solutions Legendre differential equation.

End Answer 1

Answer 2: Newtonian potentail is a solution of Laplace equation, and thus each term in power series of and thus is also solution
spherical harmonics (solution of Laplace equation in a sphere)

End Answer 2

2nd set of homogeneous solutions (non-polynomial) to Legendre equation.

1st set of homogeneous solutions (polynomial) to Legendre equation.

Legendre functions=

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HW4 page19-1 HW