User:Egm6321.f11.team3/Hwk7

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R*7.1 - Finding ds and Laplace Operator Equivalent in Spherical Coordinates[edit]

From the lecture slide Mtg 39-1 [1]

Given[edit]

Infinitesimal length in Cartesian coordinates given as in Mtg 38-5 [2];

(7.1.1)

(7.1.2)


(7.1.3)

Delta function defined as ;

(7.1.4)

(7.1.5)

(7.1.6)

(7.1.7)

(7.1.8)

(7.1.9)


Find[edit]

  • Show that infinitesimal length 'ds' can be written as (5) in meeting 39-1 in spherical coordinates.


  • Derive Laplace operator in spherical coordinates.


Solution[edit]

(7.1.10)

(7.1.11)

(7.1.12)



(7.1.13)

(7.1.14)

(7.1.15)

(7.1.16)

(7.1.17)

(7.1.18)

(7.1.19)

(7.1.20)

If we group things together we will get ;




(7.1.21)

(7.1.22)

(7.1.23)

If we substitute in

(7.1.24)




R*7.2 - Heat Conduction on a Cylinder[edit]

From the lecture slide Mtg 40-6 [3]

Given[edit]

Coordinate equivalents given as in the lecture;

(7.2.1)


Find[edit]

  • Find

(7.2.2)

  • Find

(7.2.3)

  • Find in cylindrical coordinates.
  • Use separation of variable to find the separated equations and compare to the bessel eq. (1) [4]

Solution[edit]

(7.2.4)

(7.2.5)

(7.2.6)


(7.2.7)




(7.2.8)

Equation yields

(7.2.9)


(7.2.10)

(7.2.11)

(7.2.12)

(7.2.13)

(7.2.14)

If we substitute in we will get






(7.2.15)

(7.2.16)

Dividing through

(7.2.17)

(7.2.18)

(7.2.19)

(7.2.20)

(7.2.21)

(7.2.22)

(7.2.23)

(7.2.24)

If we change we will get bessel's equation




(7.2.25)

(7.2.26)

(7.2.27)

(7.2.28)

This transformation will lead us to find the bessel equation shown in meeting 27-1.

R*7.3 Find in spherical coordinates[edit]


Given[edit]


(7.3.1)



Find[edit]


Find in spherical coordinates using the math/physics convention.

Solution[edit]


According to lecture note 38-6, we can get the following equation

(7.3.2)


(7.3.2)


(7.3.3)


(7.3.4)


(7.3.5)



(7.3.6)



(7.3.7)


(7.3.8)


(7.3.9)



(7.3.10)



(7.3.11)


(7.3.12)




(7.3.13)


Put equation (7.3.13) into equation (7.3.1).

(7.3.14)


R*7.4 Laplacian in elliptic coordinates[edit]


Given[edit]


(7.4.1)



Find[edit]


Verify the Laplacian in elliptic coordinates given by

(7.4.2)



Solution[edit]


According to Wikipedia [5]

(7.4.3)


where is a nonnegative real number and

(7.4.4)


(7.4.5)



(7.4.6)


(7.4.7)



(7.4.8)


(7.4.9)


According to hyperbolic trigonometric identity

(7.4.10)


Put equation (7.4.10) into (7.4.9).

(7.4.11)


(7.4.12)


Put equation (7.4.12) into (7.4.1).

(7.4.13)



R*7.5-Laplacian in parabolic coordinates[edit]


Given[edit]


(7.5.1)

Parabolic_coordinates

(7.5.2)


Find[edit]


Verify the Laplacian in parabolic corrdinates given by:

(7.5.3)


Solution[edit]


Slove by ourselves

(7.5.4)

(7.5.5)

(7.5.6)

(7.5.7)

Plunge into equation 7.5.2:

(7.5.8)

(7.5.9)

R*7.6 Verify Homogeneous Solution of Legendre Equation[edit]


Given[edit]


(7.6.1)


(7.6.2)


(7.6.3)


(7.6.4)


(7.6.5)


(7.6.6)


(7.6.7)


Find[edit]


Verify that 7.6.1 thru 7.6.4 are homogeneous solutions of Legendre equation 7.6.6. Show that 7.6.5 can also be written as 7.6.7. Verify that 7.6.1 through 7.6.4 can be obtained from 7.6.5 or 7.6.7.

Solution[edit]

We solved this problem on our own.


To verify 7.6.1 through 7.6.4 are homogeneous solutions of 7.6.6, from King we first find the first and second derivatives of each equation.






Now we plug these solutions into 7.6.6 in place of the y terms for the respective values of n.










Since all terms cancel to zero, the solutions are valid.



The equations 7.6.1 through 7.6.4 can be attained from 7.6.5 as shown below,











R*7.7-Obtain the separated equations for the Laplace equation in Parabolic coordinates[edit]


Given[edit]


(7.7.1)


Find[edit]


Find the separated equation for the Laplace equation

Solution[edit]


Slove by ourselves

(7.7.2)

(7.7.3)

Divided by u:

(7.7.4)

(7.7.5)

(7.7.6)

R*7.8 Heat Conduction on a Sphere[edit]


Find[edit]


Show that for n=0:

Plot the following Legendre polynomials and functions,





Observe the limits of and as

Solution[edit]

We solved this problem on our own and reviewed old homework afterwards for comparison.


We find the first four Legendre polynominials and functions from the Mathworld Wolfram site. They are,

















Observe that when x goes to 1, (1-x) is getting smaller. Thus,


The same situation happens when x goes -1.

We now plot each function using the following Matlab script and see that each one becomes asymptotic at +/-1.

clear all;
x=-1:.01:1;
P0=1;
P1=x;
P2=.5.*(3.*x.^2-1);
P3=.5.*(5.*x.^3-3*x);
Q0=.5*log((1+x)./(1-x));
Q1=(x/2).*log((1+x)./(1-x))-1;
Q2=(3*x.^2-1)/4.*log((1+x)./(1-x))-(3*x)/2;
Q3=(5*x.^3-3*x)/4.*log((1+x)./(1-x))-(5*x.^2)/2+(2/3);
figure(1)
plot(x,P0,'-',x,P1,'^',x,P2,'*',x,P3,'+')
legend('P0','P1','P2','P3')
figure(2)
plot(x,Q0,'-',x,Q1,'^',x,Q2,'*',x,Q3,'+')
legend('Q0','Q1','Q2','Q3')



Legendre Polynomials 1.jpg

Legendre Functions 1.jpg

R7.9: To Find vi[edit]


Given[edit]


Consider the non-orthonormal basis expressed in the orthonormal basis as follows



where:

Find[edit]


such that


Solution[edit]


 Solved on our own.


From the given data,



We need to find such that

So,

We also know that

So,



Hence,

and

Therefore,

References[edit]

  1. Lecture slide Mtg 39-1 [1]
  2. Lecture slide Mtg 38-5 [2]
  3. Lecture slide Mtg 40-6 [3]
  4. Lecture slide Mtg 27-1 [4]
  5. Elliptic coordinates [5]

Contributing Members[edit]

Problem Assignments
Problem Number Lecture(Mtg) Assigned To
R*7.1 Mtg 39-1 Ismail
R*7.2 Mtg 40-5 Ismail
R*7.3 Mtg 40-6 YungSheng Chang
R*7.4 Mtg 41-1 YungSheng Chang
R*7.5 Mtg 41-2 Chao Yang
R*7.6 Mtg 41-5 Robert Anderson
R*7.7 Mtg 41-6 Chao Yang
R*7.8 Mtg 42-2 Robert Anderson
R7.9 Mtg 42-6 Prashant Gopichandran
Table of Contributions
Name Problems Solved Problems Checked Signature
Robert Anderson R*7.6,R*7.8 Robert Anderson 06:15, 29 November 2011 (UTC)
YungSheng Chang R*7.3,R*7.4 R*7.8 YungSheng Chang 22:08, 21 November 2011 (UTC)


Chao Yang R*7.5,R*7.7 R*7.6 Chao Yang 00:08, 28 November 2011 (UTC)


Prashant Gopichandran R7.9 Prashant Gopichandran 18:41, 7 December 2011 (UTC)
Ismail R*7.1, R*7.2 R*7.5,R*7.7 Ismail 23:39, 6 December 2011 (UTC)