User:Egm6321.f12/Lecture plan

= All versions =

Fall 2012, Fall 2011, Fall 2010, Fall 2009

= Recorded lectures =

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.

= Lecture notes, report table =

Additional presentations (video, wiki, static html) made in class may not be recorded on these transparencies.

A mtg number n followed by a version letter in parentheses ([a-z]) indicates that this pdf file had been updated with a 2nd version "b", 3rd version "c", etc. The second letter "x" after a version letter designates an extra lecture; this letter became "m" after the extra lecture had become a make-up lecture. 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 viewer DjView, or Sumatra PDF, or evince )

Log into the course Sakai site to access the doc "PEA1 F12: Lecture notes, videos".

$$\displaystyle \bullet$$ Mtg 1, Thu, 2012.08.23, [[media:pea1.f12.sec0.djvu|sec.0]], $$\displaystyle \bullet$$ Mtg 2, Thu, 2012.08.23, [[media:pea1.f12.sec0.djvu|sec.0]], [[media:pea1.f12.sec1a.djvu|sec.1a]], $$\displaystyle \bullet$$ Mtg 3, Tue, 2012.08.28, [[media:pea1.f12.sec1a.djvu|sec.1a]], [[media:pea1.f12.sec1b.djvu|sec.1b]], $$\displaystyle \bullet$$ Mtg 4, Thu, 2012.08.30, [[media:pea1.f12.sec1b.djvu|sec.1b]], [[media:pea1.f12.sec4.djvu|sec.4]], $$\displaystyle \bullet$$ Mtg 5, Thu, 2012.08.30, [[media:pea1.f12.sec4.djvu|sec.4]], $$\displaystyle \bullet$$ Mtg 6, Tue, 2012.09.04, [[media:pea1.f12.sec4.djvu|sec.4]], [[media:pea1.f12.sec5.djvu|sec.5]], [[media:pea1.f12.sec6.djvu|sec.6]], $$\displaystyle \bullet$$ Mtg 7, Thu, 2012.09.06, [[media:pea1.f12.sec5.djvu|sec.5 (b)]], [[media:pea1.f12.sec6.djvu|sec.6 (b)]], $$\displaystyle \bullet$$ Mtg 8, Thu, 2012.09.06, [[media:pea1.f12.sec7.djvu|sec.7]], [[media:pea1.f12.sec2a.djvu|sec.2a]], $$\displaystyle \bullet$$ Mtg 9 (x), Sat, 2012.09.08, [[media:pea1.f12.sec7.djvu|sec.7]], $$\displaystyle \bullet$$ Mtg 10, Tue, 2012.09.11, [[media:pea1.f12.sec4.djvu|sec.4 (c)]], [[media:pea1.f12.sec7.djvu|sec.7 (b)]], [[media:pea1.f12.sec8.djvu|sec.8]], $$\displaystyle \bullet$$ Mtg 11, Thu, 2012.09.13, [[media:pea1.f12.sec7.djvu|sec.7 (c)]], [[media:pea1.f12.sec8.djvu|sec.8 (b)]], [[media:pea1.f12.sec9.djvu|sec.9]], $$\displaystyle \bullet$$ Mtg 12, Thu, 2012.09.13, [[media:pea1.f12.sec10.djvu|sec.10]], $$\displaystyle \bullet$$ Mtg 13, Fri, 2012.09.14, [[media:pea1.f12.sec11.djvu|sec.11]], $$\displaystyle \bullet$$ Mtg 14, Tue, 2012.09.18, [[media:pea1.f12.sec0.djvu|sec.0 (b)]], [[media:pea1.f12.sec8.djvu|sec.8 (c)]], [[media:pea1.f12.sec12.djvu|sec.12]], $$\displaystyle \bullet$$ Mtg 15, Thu, 2012.09.20, [[media:pea1.f12.sec13.djvu|sec.13]], $$\displaystyle \bullet$$ Mtg 16, Thu, 2012.09.20, [[media:pea1.f12.sec14.djvu|sec.14]], $$\displaystyle \bullet$$ Mtg 17, Tue, 2012.09.25, [[media:pea1.f12.sec15.djvu|sec.15]], $$\displaystyle \bullet$$ Mtg 18, Wed, 2012.09.26, [[media:pea1.f12.sec16.djvu|sec.16]], $$\displaystyle \bullet$$ Mtg 19, Thu, 2012.09.27, [[media:pea1.f12.sec9.djvu|sec.9 (b)]], [[media:pea1.f12.sec17.djvu|sec.17]], $$\displaystyle \bullet$$ Mtg 20, Tue, 2012.10.02, [[media:pea1.f12.sec18.djvu|sec.18]], $$\displaystyle \bullet$$ Mtg 21, Thu, 2012.10.04, [[media:pea1.f12.sec18.djvu|sec.18]], [[media:pea1.f12.sec21.djvu|sec.21]], $$\displaystyle \bullet$$ Mtg 22, Thu, 2012.10.04, [[media:pea1.f12.sec21.djvu|sec.21]], [[media:pea1.f12.sec22.djvu|sec.22]], $$\displaystyle \bullet$$ Mtg 23, Tue, 2012.10.09, [[media:pea1.f12.sec15.djvu|sec.15]], [[media:pea1.f12.sec18.djvu|sec.18]], [[media:pea1.f12.sec21.djvu|sec.21]], [[media:pea1.f12.sec24.djvu|sec.24]], $$\displaystyle \bullet$$ Mtgs 24-25, Thu, 2012.10.11, Exam 1, $$\displaystyle \bullet$$ Mtg 26, Sun, 2012.10.14, [[media:pea1.f12.sec20.djvu|sec.20]], $$\displaystyle \bullet$$ Mtg 27, Tue, 2012.10.16, [[media:pea1.f12.sec63.djvu|sec.63]], $$\displaystyle \bullet$$ Mtg 28, Thu, 2012.10.18, [[media:pea1.f12.sec21.djvu|sec.21 (b)]], [[media:pea1.f12.sec22.djvu|sec.22 (b)]], $$\displaystyle \bullet$$ Mtg 29, Thu, 2012.10.18, [[media:pea1.f12.sec22.djvu|sec.22 (b)]], $$\displaystyle \bullet$$ Mtg 30, Fri, 2012.10.19, [[media:pea1.f12.sec64.djvu|sec.64]], $$\displaystyle \bullet$$ Mtg 31, Tue, 2012.10.23, [[media:pea1.f12.sec27.djvu|sec.27]], $$\displaystyle \bullet$$ Mtg 32, Thu, 2012.10.25, [[media:pea1.f12.sec27.djvu|sec.27.(b)]], [[media:pea1.f12.sec30.djvu|sec.30]], $$\displaystyle \bullet$$ Mtg 33, Thu, 2012.10.25, [[media:pea1.f12.sec30.djvu|sec.30]], $$\displaystyle \bullet$$ Mtg 34, Wed, 2012.10.31, [[media:pea1.f12.sec30.djvu|sec.30.(b)]], [[media:pea1.f12.sec31.djvu|sec.31]], $$\displaystyle \bullet$$ Mtg 35, Thu, 2012.11.01, [[media:pea1.f12.sec32.djvu|sec.32]], $$\displaystyle \bullet$$ Mtg 36, Thu, 2012.11.01, [[media:pea1.f12.sec33.djvu|sec.33]], $$\displaystyle \bullet$$ Mtg 37, Sat, 2012.11.03, [[media:pea1.f12.sec34.djvu|sec.34]], $$\displaystyle \bullet$$ Mtg 38, Tue, 2012.11.06, [[media:pea1.f12.sec33.djvu|sec.33.(b)]], [[media:pea1.f12.sec35.djvu|sec.35]], [[media:pea1.f12.sec64.djvu|sec.64.(b)]], $$\displaystyle \bullet$$ Mtg 39, Thu, 2012.11.08, [[media:pea1.f12.sec36.djvu|sec.36]], $$\displaystyle \bullet$$ Mtg 40, Thu, 2012.11.08, [[media:pea1.f12.sec37.djvu|sec.37]], $$\displaystyle \bullet$$ Mtg 41, Sat, 2012.11.11, [[media:pea1.f12.sec64.djvu|sec.64.(c)]], $$\displaystyle \bullet$$ Mtg 42, Tue, 2012.11.13, [[media:pea1.f12.sec33.djvu|sec.33.(c)]], [[media:pea1.f12.sec64.djvu|sec.64.(d)]], [[media:pea1.f12.sec37.djvu|sec.37]], [[media:pea1.f12.sec38.djvu|sec.38]], $$\displaystyle \bullet$$ Mtg 43, Thu, 2012.11.15, [[media:pea1.f12.sec38.djvu|sec.38]], [[media:pea1.f12.sec39.djvu|sec.39]], $$\displaystyle \bullet$$ Mtg 44, Fri, 2012.11.16, [[media:pea1.f12.sec39.djvu|sec.39.(b)]], $$\displaystyle \bullet$$ Mtg 45, Tue, 2012.11.20, [[media:pea1.f12.sec40.djvu|sec.40]], [[media:pea1.f12.sec64.djvu|sec.64.(e)]],

[[media: pea1.f11.my.scratch.at.airport.pdf | My scratch at an airport]] for the hint on [[media:pea1.f12.sec22.djvu| p.22-4]].

Report table

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

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

$$\displaystyle \clubsuit$$ Kailath, T., Linear systems, Prentice Hall, 1980. UF library QA402 .K295 1980 google amazon

$$\displaystyle \clubsuit$$ 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

$$\displaystyle \clubsuit$$ 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

$$\displaystyle \clubsuit$$ 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.

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

MIT OpenCourseWare's past and future. R. Suehle, OpenSource.com, 8 Sep 2011.

$$\displaystyle \spadesuit$$ 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."

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

WolframAlpha syntax.

$$\displaystyle \spadesuit$$ Stephen Wolfram: Computing the theory of everything, video on TED.com

$$\displaystyle \spadesuit$$ 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.

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

SPECFUN - Special Function Evaluation

Open-source software and documentation
$$\displaystyle \spadesuit$$ 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)

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

 NEW: 

S. Bodenmann The 18th-century battle over lunar motion, Physics Today, Jan 2010.

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

$$\displaystyle \clubsuit$$ 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)

$$\displaystyle \clubsuit$$ 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 =

$$\displaystyle \clubsuit$$ 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.

$$\displaystyle \clubsuit$$ 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.

$$\displaystyle \clubsuit$$ 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 2011)
$$\displaystyle \color{red}{\clubsuit}$$

 NEW: 

How Not to Plan for the Future, New York Times, Editorial, April 20, 2011. See also New York Times: High-speed rail deserves continued support, Transportation for America, 2011.04.21.

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

 NEW: 

High-Speed Rail Poised to Alter China, New York Times, By KEITH BRADSHER Published: June 22, 2011.

News on high-speed train (Aug 2010)
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.

Solving PDEs in curvilinear coordinates




= 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.

Helmholtz equation (PDE)
Helmholtz equation (Wikipedia)

Reduction of order method 0: Missing dependent variable
Zwillinger (1998), Sec 55

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).

10 Best Soccer Goals Ever (2nd best by Roberto Carlos in Brazil vs France)

Best corner kick ever!

NOVA: Bombing Hitler's dam: 24:00 to 29:00

The Knuckleball: The science behind why it's so damn hard to hit

Accuracy in paintball

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.

Roll control of rocket


Bryson & Ho 1975, p.169



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)

Connection of linear systems with Reynolds Transport Theorem





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

Legendre L2-ODE-VC (2)
King et al. (2003), p.31

Nonlinear nth-order ODEs
Zwillinger (1998), Sec 63, p.289

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

Condition 2: Single exactness relation
involving $$f_i := \partial F / \partial y^{(i)} $$.

Vibration: Chladni patterns
Chladni figures Chladni patterns (youtube) Chladni patterns on a vibrating plate excited by an acoustic speaker

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

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.

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

Non-homogeneous L2-ODE-VC
Zwillinger (1998), Sec 94

Non-homogeneous L2-ODE-CC
Boyce & DiPrima

Variation of parameters
Gravitational perturbation (wikipedia)



Big Picture, 2010 (video, Hulu): "What we see from space is the big picture of Earth. This is a science documentary with a difference". NASA GRACE satellite measures earth mass variations; time markers on video for this story: 34 min - 40 min. google search for "nasa grace satellite"

Communicating the science of climate change, Physics Today, Oct 2011, p.48.

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

Tacoma Narrows Bridge collapse
Puget sound (NASA satellite image)

= Legendre functions =

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

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

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

Bessel differential equation (1)
King et al. (2003), p.80

elliptic coordinates (2D)
Elliptic coordinates (wikipedia)



Parabolic coordinates
Parabolic 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.

Legendre polynomials
Legendre polynomials (wikipedia)

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

$$\psi (r,\theta) = \sum_{n} \left(A_n r^n + B_n r^{-(n+1)} \right ) \left[ C_n P_n (\mu) + D_n Q_n (\mu) \right ] \, \quad \mu := \sin \theta$$

Binomial theorem
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.

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)

Bypassing 2nd homogeneous solution
King et al. (2003), p.44

Rodrigues's formula
Olinde Rodrigues (1795-1851)

 Lecture plan of other courses 