User:Egm6321.f10/Lecture plan

Under construction throughout the semester. Egm6321.f10 22:15, 25 August 2010 (UTC)

if I take the trouble of writing the dream down, when I awaken the next morning I can remember the dream perfectly well without referring to my notes. The same thing is true of, for example, remembering a telephone number. If I am told a number and merely think about it, I am likely to forget it or transpose some of the digits. If I repeat the numbers out loud or write them down, I can remember them quite well. This surely means that there is a part of our brain which remembers sounds and images, but not thoughts. Carl Sagan, The Dragons of Eden, 1977 (Pulitzer Prize).

there is general agreement that most current [science] standards lead to curricula and textbooks that contain too many topics covered in too little depth. [Conceptual Framework for New Science Education Standards], [Frequently asked questions], National Academies, 2010.

Give me a fish, I eat for one day. Teach me to fish, I eat for a lifetime.

Proverb quoted in [Lebesgue integration, S.B. Chae, 1995].

All versions

Fall 2011, Fall 2010, Fall 2009

Recorded lectures, TA user page

Recorded lectures in E-Learning at UF (password required): Click Continue, type username and password, click at EGM 6321, click Course Content; at "Lecture Videos" link, click at drop down menu, then select "Preview" to go to the web page with lecture video links.

TA user page: Summary of HW statements, for students to interact with TA.

Lecture notes, report table

These lecture notes were mostly written in real time during the lectures (i.e., not prepared ahead of the lectures, except for a few extra lectures). Additional presentations (video, wiki, static html) made in class may not be recorded on these transparencies.

A mtg number n followed by a lowercase alphabet in parentheses ([a-z]) indicates that this pdf file had been updated with a 2nd version "b", 3rd version "c", etc. If you had downloaded this pdf file before, you want to clear the cache of your browser so to get the new version.

djvu: (install the viewers evince or DjView4)
Mtg 1 (c), Mtg 2 (d), Mtg 3 (b), Mtg 4 (c), Mtg 5 (b), Mtg 6 (d), Mtg 7 (c), Mtg 8, Mtg 9 (b), Mtg 10 (b), Mtg 11, Mtg 12, Mtg 13 (c), Mtg 14 (b), Mtg 15 (b), Mtg 16, Mtg 17 (b), Mtg 18, Mtg 19, Mtgs 20+21: Exam 1, Mtg 22 (b), Mtg 23, Mtg 24 (c), Mtg 25 (b), Mtg 26 (b), Mtg 27 (b), Mtg 28 (c), Mtg 29 (c), Mtg 30 (c), Mtg 31 (b), Mtg 32 (b), Mtg 33 (c), Mtg 34 (b), Mtg 35 (b), Mtg 36 (b), Mtg 37 (b), Mtg 38, Mtg 39 (b), Mtg 40 (b), Mtg 41 (c), Mtgs 42+43: Exam 2, Mtg 44 (b), Mtg 45 (c), Mtg 46 (b),

Mediawiki transcripts: Mtg 1, Mtg 2, Mtg 3, Mtg 4, Mtg 5, Mtg 6, Mtg 7, Mtg 8, Mtg 9, Mtg 10, Mtg 11, Mtg 12, Mtg 13, Mtg 14, Mtg 15, Mtg 16, Mtg 17, Mtg 18, Mtg 19, Mtg 20, Mtg 21, Mtg 22, Mtg 23, Mtg 24, Mtg 25, Mtg 26, Mtg 27, Mtg 28, Mtg 29, Mtg 30, Mtg 31, Mtg 32, Mtg 33, Mtg 34, Mtg 35, Mtg 36, Mtg 37, Mtg 38, Mtg 39, Mtg 40, Mtg 41, Mtg 42, Mtg 43, Mtg 44, Mtg 45, Mtg 46,

Report table

$\displaystyle \color {red}{\clubsuit }$ NEW:

Inspiring video on teamwork and learning: A hole in the wall: How children learn without a teacher

Mourning the Death of Handwriting, By Claire Suddath. Time Magazine, Monday, Aug. 03, 2009.

Op-Art: The Write Stuff, by Inga Dubay and Barbara Getty, NY Times, 8 Sep 2009.

References

Books

$\displaystyle \clubsuit$ A.C. King, J. Billingham, S.R. Otto, Differential equations: Linear, nonlinear, ordinary, partial, Cambridge University Press, 2003. ISBN-10: 0521016878 ISBN-13: 978-0521016872. UF library 0511078315 (electronic bk.) Google books Amazon.com

$\displaystyle \clubsuit$ D. Zwillinger, Handbook of Differential Equations, Third Edition, Academic Press, 1998. ISBN-10: 0127843965. ISBN-13: 978-0127843964. UF library QA371.Z88 1989, 2 copies, one for in-library use. Google books Amazon.com

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Bryson, A.E., and Ho, Y.C., Applied optimal control, Taylor & Francis, 1975. UF library QA402.3 .B78 1969 google amazon

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Kailath, T., Linear systems, Prentice Hall, 1980. UF library QA402 .K295 1980 google amazon

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Kline, M., Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972. UF library QA21.K516 [Vol.1 google] Vol.2 google Vol.3 google Vol.1 amazon Vol.2 amazon Vol.3 amazon

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Boyce, W.E., DiPrima, R.C., Elementary Differential Equations and Boundary Value Problems, 7th edition, Wiley, 2001. UF library QA371 .B773 1986 amazon

$\displaystyle \clubsuit$ O.D. Kellogg, Foundations of potential theory, Dover publications, 1954. UF library QA825 .K4x 1953 Google books Amazon.com

$\displaystyle \clubsuit$ P.M. Morse, H. Feshbach, Methods of theoretical physics, Parts I & II, McGraw-Hill, 1953. UF library QC20 .M6 google books amazon.com

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NIST Digital Library of Mathematical Functions, companion of the NIST Handbook of Mathematical Functions, ed. by F.W.J. Olver, et al., Cambridge U. Press, 2010. "Together these works represent a successor to the highly successful Handbook of Mathematical Functions (M. Abramowitz and I. Stegun, Eds.; see below), which was published by the National Bureau of Standards in 1964." NA Digest, Vol.10, No.19, May 2010.

$\displaystyle \clubsuit$ M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia

$\displaystyle \clubsuit$ N.N. Lebedev, Special Functions & Their Applications, Dover Publications, 1972. ISBN 0486606244 (pbk). UF library QA351.L3613 1972 Google books Amazon.com

$\displaystyle \clubsuit$ K. Oldham, J. Myland, J. Spanier, An atlas of functions, 2nd edition, Springer, 2008. 1st edition, UF library QA331.S685 1987 google books amazon.com

$\displaystyle \clubsuit$ A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, 1991. UF library QC20.7.O75N5513 1991 Google Amazon

Web references

$\displaystyle \spadesuit$ Related MIT OpenCourseWare courses

$\displaystyle \spadesuit$ Wolfram|Alpha is ... "learning resource available to your students at no cost that works as a computational knowledge engine. Wolfram|Alpha is not a search engine like Google or Yahoo!, because unlike a traditional search engine, Wolfram|Alpha has the capability to instantly compute the answer to previously unasked questions instead of scouring the web and returning links to pages that already exist. The results are displayed in an easy-to-read, understandable format that can be used as a primary source for educational and academic purposes."

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Stephen Wolfram: Computing the theory of everything, video on TED.com

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Wolfram online integrator

$\displaystyle \spadesuit$ EqWorld, The World of Mathematical Equations. It is a good idea to verify the sources, as the site is not responsible for accuracy and correctness; see Rights and obligations of contributors and website administration.

• 2nd-order linear ODEs
• 2nd-order non-linear ODEs
• ODE education

$\displaystyle \spadesuit$ ODE (Wikipedia): Be careful; always verify the sources.

Open-source software and documentation

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Symbolic computation

Papers

$\displaystyle \clubsuit$ Vu-Quoc, L., and Olsson, M., Formulation of a basic building-block model for interaction of high-speed vehicles on flexible structures, ASME Journal of Applied Mechanics, Vol.56, No.2, pp.451-458, 1989. (pdf)

$\displaystyle \clubsuit$ Vu-Quoc, L., and Olsson, M., A computational procedure for interaction of high-speed vehicles on flexible structures without assuming known vehicle nominal motion, Computer Methods in Applied Mechanics and Engineering, Vol.76, pp.207-244, 1989. (pdf)

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NEW:

Goff, J.E., Power and spin in the beautiful game, Physics Today, Vol.63, No.7, pp.62-63, Jul 2010.

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NEW:

Irwin, P.A., Vortices and tall buildings: A recipe for resonance Physics Today, Vol.63, No.9, pp.68-69, Sep 2010. Simulation of vortex formation and shedding.mpg The Burj Khalifa building in Dubai (wikipedia)

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NEW:

Narasimhan, T.N., Thermal conductivity through the 19th century Physics Today, Vol.63, No.8, pp.36-41, Aug 2010.

$\displaystyle \clubsuit$ Vu-Quoc, L., and Tran, V.X., Singularity analysis and fracture energy release rate for composites: Piecewise homogeneous-anisotropic materials, Computer Methods in Applied Mechanics and Engineering, John H. Argyris Memorial Issue, Vol.195, No.37-40, pp.5162-5197, 15 July 2006. (pdf)

Inspiring engineering news

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NEW:

Nova Documentary Examines Science Behind Chilean Miners' Rescue, PBS News Hour, AIR DATE: Oct. 26, 2010. In a documentary airing on most PBS stations, "Nova" captured the gripping story of the 33 Chilean miners trapped underground for more than two months and explores the science used to rescue them.

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NEW:

Chilean Miners' Rescue Presented Massive Engineering, Drilling Hurdle, PBS News Hour, AIR DATE: Oct. 26, 2010. Three separate drilling plans were created to save the 33 Chilean miners trapped in the San Jose mine. Margaret Warner talks to one Americans who helped orchestrate the rescue.

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NEW:

NOVA: Emergency Mine Rescue, Engineers and NASA scientists aid an all-out effort to save 33 Chilean miners trapped nearly half a mile underground. Aired October 26, 2010 on PBS.

Motivation

High-speed maglev trains

$\displaystyle \clubsuit$ German Transrapid (Maglev), electromagnetic (attraction) maglevs. German Transrapid Emsland 500 km/hr (video, 7:37 min) (500 / 1.6 = 312.5 mi / hr). The Transrapid story (video, history of development, electromagnetic systems vs electrodynamic systems): Part 1 (8:28 min) Part 2 (8:07 min). Shanghai maglev train (video, 5:35 min).

Equations of motion for high-speed vehicles interacting with flexible guideways; see Vu-Quoc & Olsson (1989 a b ): Actually, system of coupled nonlinear 2nd-order ordinary differential equations and partial differential equations.

$\displaystyle \clubsuit$ Japanese Maglev, electrodynamic (repulsion) maglevs, with retractable wheels. Japanese maglev at 581 km/hr (video, 4:43 min) (581 / 1.6 = 363 mi / hr).

$\displaystyle \clubsuit$ French TGV speed record 574.8 km / hr = 357.2 mi / hr (non maglev, wheel-on-rail trains)

Recent news on high-speed train (Aug 2010)

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NEW:

Fla. seeks $1B more to build bullet train, Miami Herald, 28 Aug 2010.

Debate on high-speed rail (Sep 2009)

Siemens Fills Russia’s Need for High-Speed Train, by A.E. Kramer, NY Times, September 24, 2009.

Stimulus Puts High-Speed Rail On The Fast Track, NPR, Morning Edition, 24 Feb 2009; audio 4:10 min.

States Make Pitches For High-Speed-Rail Money, NPR, All Things Considered, 21 Aug 09, audio 7:42 min.

California Edges Ahead In High-Speed-Train 'Race', NPR, All Things Considered, 3 Sep 09.

High-Speed Rail Skeptic Outlines Position, NPR, All Things Considered, 3 Sep 09.

More with Search for "npr high speed rail train " Search for "npr high speed train florida"

Equations of motion

Coupled nonlinear 2nd-order ODE and PDEs with varying coefficients, which depend on the unknown functions to be solved for. Particularization to linear 2nd-order ODEs with varying coefficients.

Linear 2nd-order ODEs with varying coefficients (L2-ODE-VC)

Even though this section is about linear 2nd-order ODEs with varying coefficients, many of the methods listed in this section apply to nonlinear ODEs in general, with linear ODEs as particular cases. We present the general nonlinear case first, then particularize to the linear case.

Applications: Homogeneous L2-ODE-VC

Helmholtz equation (PDE)

Helmholtz equation (Wikipedia)

Reduction of order method 0: Missing dependent variable

Zwillinger (1998), Sec 55

Nonlinear 2nd-order ODEs (N2-ODEs)

Integrating-factor method (1)

Zwillinger (1998), Sec 79

Application: Motion of rocket with air resistance

See also Kline, 1972, p.476: John Bernoulli's equation; contributions of Euler (1734-35) and Clairaut (1739-40) to exactness condition and integrating factor method (IFM).

L2-ODE-VC with missing dependent variable

General L1-ODE-VC (2)

See General L1-ODE-VC (1).

Control engineering: Linear systems

Bryson & Ho 1975, p.450

Kailath 1980, p.165, Eq.(21); Sec. 2.5, p.160, Solutions of state equation and modal decomposition; Sec 2.5.1. Time invariant equations and matrix exponentials.

Irwin, P.A., Vortices and tall buildings: A recipe for resonance Physics Today, Vol.63, No.9, pp.68-69, Sep 2010. Simulation of vortex formation and shedding.mpg The Burj Khalifa building in Dubai (wikipedia)

Reduction of order method 1: Exact nonlinear ODEs

Zwillinger (1998), Sec 63: Applicable to nonlinear ODEs, in particular linear 2nd-order ODEs.

Nonlinear 1st-order ODEs (2)

See Nonlinear 1st-order ODEs (N1-ODEs) (1) and Integrating-factor method: Nonlinear 1st-order ODEs

Nonlinear 2nd-order ODEs

Legendre L2-ODE-VC (2)

King et al. (2003), p.31

Nonlinear nth-order ODEs

Zwillinger (1998), Sec 63, p.289

General Nn-ODEs

$F(x,y^{(0)},y^{(1)},\ldots ,y^{(n)})=0$

Two exactness conditions

Condition 2: Single exactness relation

involving $f_{i}:=\partial F/\partial y^{(i)}$.

Special case: Euler equation

Application: Engineering

Vibration: Chladni patterns

One of Chladni's best-known achievements was inventing a technique to show the various modes of vibration on a mechanical surface.

... Since the 20th century it has become more common to place a loudspeaker driven by an electronic signal generator over or under the plate to achieve a more accurate adjustable frequency. In Ernst Chladni (1756-1827).

Chladni figures Chladni patterns (youtube) Chladni patterns on a vibrating plate excited by an acoustic speaker Vibrating modes of a guitar plate

Acoustics of drums, by T.D. Rossling, Physics Today, Vol.45, No.3, pp.40-47, Mar 1992.

Generalization: Euler (homogeneous) Ln-ODE-VC (1)

Zwillinger (1998), Sec 61

Special case: Non-homogeneous Ln-ODE-CC

Non-homogeneous L2-ODE-CC

Table of particular solutions

Boyce & DiPrima 2001, p.175.

Superposition of solution for L2-ODE-VC

Here, we treat the cases in which at least a solution (homogeneous or particular) can be guessed by inspection; the other solution can then be generated from the guessed solution.

For the cases in which the solutions cannot be guessed by inspection, see Solution by power series: Frobenius method.

Homogeneous (complementary) solution

Euler (homogeneous) Ln-ODE-VC (2)

See above.

Zwillinger (1998), Sec 61

Method of trial solution (undetermined coefficients)

This method is also known as the method of undertermined coefficients; the terminology "trial solution" is more descriptive since the method involves guessing the solution mathematical expression, called the trial solution, which have unknown coefficients to be determined by substituting the trial solutions into the differential equation.

King et al. (2003), Appendix 5

Zwillinger (1998), Sec 94

Wikipedia

Reduction of order method 2: Undetermined factor (1)

King et al. (2003), p.5

Zwillinger (1998), Sec 85

Particular solution

Method of trial solution

Non-homogeneous L2-ODE-VC

Zwillinger (1998), Sec 94

Non-homogeneous L2-ODE-CC

Boyce & DiPrima

Variation of parameters

King et al. (2003), p.7 Zwillinger (1998), Sec 95

Legendre functions

Kellogg (1953), p.125 King et al. (2003), p.31

Motivation: Heat conduction

From 1807 to 1811, Joseph Fourier conducted experiments and devised mathematical techniques that together yielded the first estimate of a material's thermal conductivity. His methodology has influenced all subsequent work. ... In 1802, upon his return to France from Napoleon’s Egyptian campaign, Fourier was appointed prefect of the department of Isère. Despite heavy administrative responsibilities, Fourier found time to study heat diffusion. He was inspired by deep curiosity about Earth and such phenomena as the attenuation of seasonal temperature variations in Earth’s subsurface, oceanic and atmospheric circulation driven by solar heat, and the background temperature of deep space. Fourier began with a paper by Jean Baptiste Biot (1774–1862). ... [who] had attempted to formulate a differential equation for heat conduction in a rod heated at one end and able to dissipate heat to the atmosphere... based on action at a distance and Newton’s law of cooling, which states that the rate of cooling of an object is proportional to the difference between the object’s temperature and that of the atmosphere. That line of attack was unsuccessful because Newton’s law, appropriate for radiative heat loss, is inadequate for conductive transfer. Starting with Biot’s approach, Fourier obtained mathematical results that were incorrect and unsatisfactory. He then abandoned the action-at-a-distance approach and, based on his own physical reasoning, concluded that temperature varied continuously along the length of the rod. ... not satisfied with the 1807 work. It took him an additional three years to go beyond the discrete finite-difference description of flow between constant-temperature surfaces and to express heat flux across an infinitesimally thin surface segment in terms of a temperature gradient. Narasimhan, Physics Today, Aug 2010.

Legendre ... studied the attraction of ellipsoids. He gave a proof of a result due to Maclaurin, that the attractions at an external point lying on the principal axis of two confocal ellipsoids was proportional to their masses. He then introduced what we call today the Legendre functions and used these to determine, using power series, the attraction of an ellipsoid at any exterior point. Legendre submitted his results to the Académie des Sciences in Paris in January 1783 and these were highly praised by Laplace in his report delivered to the Académie in March. Adrien-Marie Legendre (1752-1833)

Orthogonal coordinates

Orthogonal coordinates (wikipedia): See the table of different orthogonal coordinate systems at the bottom of the article.

Spherical coordinates

Axisymmetric solution

Legendre polynomials (1)

On the Law of cosines and its history: "Al-Battani was ... born in Harran, in 850. Al-Battani made his remarkably accurate astronomical observations. ... described as a famous observer and geometer. In his astronomical work he gave his own observations of the sun, moon, and the planets, more accurately than what is found in Ptolemy’s Almagest. Al-Battani’s most important book is Kitab al-Zij. It begins with the necessary mathematical tools, such as the sexagesimal numeral system and the trigonometric functions. ... in his trigonometry al-Battani worked with what we call the cosine, the sine of the complementary angle. Al-Battani’s work became important to later European astronomers like Copernicus, Brahe, Kepler and Galileo. In fact, Kepler got the idea to the Law of Cosines from al-Battani [Holme, Geometry, 2010, pp.188-89.]"

Al-Khwārizmī and his colleagues worked at the [House of Wisdom in Baghdad]. They started a magnificent scientific tradition which lasted well into the fifteenth century, when the tradition was continued by the Europeans. Today many historians of mathematics realize that the Arabs have not been given their due credit for the significant contributions they made to mathematics. The Arabs had been seen merely as preservers, commentators and "messengers", who delivered ancient Greek mathematics to the proper heirs so to speak, namely the Europeans. But today the general feeling is that this view is unjustified, since Arabic mathematicians made very significant and original contributions. It is no accident that the word algebra is derived from the title of one of al-Khwarizmi’s fundamental books, Al-kitab al-muhtasar fi hisab al-jabr wa-l-al-muqabala, abbreviated to Hisab al-jabr wa-l-al-muqabala. This is the first book to be written on algebra as such. The title means something like The condensed book on arithmetic by "aljabr" and "al-muqabala", the two Arabic words meaning, respectively, "setting together" and "balancing." The first word is the origin of our algebra. It is told that in southern Spain barbers used to be called algebraists, presumably because their duties included performing simple surgical procedures such as reducing a fracture. [Holme, Geometry, 2010, p.181.]

Northern Italy in the early thirteenth century was a land subdiv­ided into multiple feuding city-states. Among the many remnants

of the defunct Roman Empire was a numerical system (i, ii, iii, iv . . . ) singularly ill-suited to complex mathematical calculation, let alone the needs of commerce. ... By comparison, economic life in the Eastern world - in the Abassid caliphate or in Sung China - was far more advanced ... To discover modern finance, Europe needed to import it. In this, a crucial role was played by a young mathematician called Leonardo of Pisa, or Fibonacci ... The son of a Pisan customs official based in what is now Bejaia in Algeria, the young Fibonacci had immersed himself in what he called the 'Indian method' of mathematics, a combination of Indian and Arab insights. His introduction of these ideas was to revolutionize the way Europeans counted. Nowadays he is best remembered for the Fibonacci sequence of numbers ... Most important of all was Fibonacci's introduction of Hindu-Arabic numerals. He not only gave Europe the decimal system, which makes all kinds of calculation far easier than with Roman numerals; he also showed how it could be applied to commercial bookkeeping, to currency conversions and, crucially, to the cal­culation of interest. [Ferguson, The ascent of money, 2008, pp.31-32.]

Coefficients of series solution

Lecture transparency p.33-1, Eq.(2) and Eq.(5):

$\displaystyle \langle f,P_{m}\rangle =\int \limits _{\theta =-\pi /2}^{\theta =\pi /2}f(\theta )P_{m}(\sin \theta )d(\sin \theta )=\int \limits _{\mu =-1}^{\mu =1}f(\mu )P_{m}(\mu )d(\mu )$

Integration involving transcendental functions: Abramovitz & Stegun, p.77

Circular cylinder (cylindrical) coordinates

Separation of variables (3)

Bessel differential equation (1)

King et al. (2003), p.80

elliptic coordinates (2D)

Elliptic coordinates (wikipedia)

Application: Gauss-Legendre quadrature

Gauss quadrature (wikipedia) Legendre polynomials (wikipedia) Abramovitz & Stegun, p.887 Abramovitz & Stegun, Table 25.4, p.916

"What's the application?" lesson 2

3 Americans Share Nobel for Medicine, By Nicholas Wade, NY Times, Published: October 5, 2009.

The discoveries were made some 20 years ago in pursuit of a purely scientific problem that seemingly had no

practical relevance. But telomeres have turned out to play a role in two medical areas of vast importance, those of aging and cancer, because of their role in limiting the number of times a cell can divide. Wade (2009)

Two homogeneous solutions: Legendre functions

Legendre polynomials

Legendre polynomials (wikipedia)

General solution of Laplace equation

Axisymmetric case

Using the astronomy convention for spherical coordinates, the general solution for the Laplace equation before applying any boundary conditions is:

$\psi (r,\theta )=\sum _{n}\left(A_{n}r^{n}+B_{n}r^{-(n+1)}\right)\left[C_{n}P_{n}(\mu )+D_{n}Q_{n}(\mu )\right]\ ,\quad \mu :=\sin \theta$

Historical development

Application: Attraction of two spheres

Expansion into polynomial series

Binomial theorem

Not until eight years after James Bernoulli died in 1705 was his main work, the Ars Conjectandis, published. ... the first significant book on probability ... most important new result ... still name for Bernoulli. Specifically, if p and q are the respective probabilities that a single event will or will not occur ... the kth term in the binomial expansion $(p+q)^{n}$ ... is the probability that the event will occur exactly k times in the n trials. The trials are called the Bernoulli trials. The binonial expansion, for positive integer n, was familiar to the Arabs of the thirteenth century... [Battin, Astrodynamics, 1998, p.661.]

... the binomial coefficients organized in the pattern which we know today as the triangle of Pascal. This pattern had been explored by al-Karajial-Karaji, an eminent Arabic algebraist who lived from 953 to about 1029. ... regarded as the first mathematician who freed algebra from geometrical operations and replace them with the type of operations which are at the core of algebra today. ... Another mathematical contribution was Nasir’s manuscript, dated 1265, concerning the calculation of nth roots of an integer. This work is probably an exposition of material coming from al-Karaji’s school. In the manuscript Nasir determines the coefficients of the expansion of a binomial to any power giving the binomial formula and the “Pascal” triangle for binomial coefficients. [Holme, Geometry, 2010, p.200, p.210.]

NOTE: Battin 1998, p.661, probably referred to Nasir's work in the 13th century, but the binomial theorem was actually known to al-Karajial-Karaji 200 years earlier.

Application: Laminar flow around a sphere

Laplace equation in spherical coordinates

Fluid flow experiment around a cylinder (video) (not a sphere, but the streamlines are similar to those of a flow around a sphere)

Moving cylinder in a fluid (video)

Non-homogeneous Legendre equation

Alternative method

Bypassing 2nd homogeneous solution

King et al. (2003), p.44

Rodrigues's formula

Olinde Rodrigues (1795-1851)

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Lecture plan of other courses

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