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

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Mtg 36: Thu, 12 Nov 09


Let i.e. is a polynomial of degree


(1) page33-4

End Example

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

Trapezoidal Rule:



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.


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=


HW4 page19-1 HW