Given a normal distribution, confidence levels corresponding to 1σ and 2σ are ilustrated above. This illustration is taken from the following webpage entitled Drawing conclusions from data: Statistics don't lie--or do they?. You may find some useful tips on this page.

Table of Contents


Physics Department Links


Instructor Contact Information

Professor: Howard Haber

Office ISB 326
Phone 459-4228
Office Hours Mondays, 2--4 pm
e-mail haber@scipp.ucsc.edu


Teaching Assistant

Angelo Monteux
Office ISB 329
Phone 459-1962
Discussion Section Mondays, 5--6:10 pm in Earth & Marine Sciences Building, Room B214
e-mail amonteux@ucsc.edu


Learning Support Services Tutor

Benjamin Michlin

LSS tutoring sessions Tuesdays, 4--5 pm
meeting point S&E library at the entrance to the left, before the stairs
e-mail         bmichlin@ucsc.edu


Required textbook

Click here for the errata.


Miscellaneous

Displayed above are graphs of the five Legendre polynomials, Pn(x) over the interval 0≤x≤1 for n=1,2,3,4 and 5. These functions are orthonormal with respect to the standard inner product. For further details see Wolfram MathWorld.


Displayed above are graphs of the first five Bessel functions of the first kind, Jn(x), for n=0,1,2,3 and 4 over the range 0 ≤ x ≤ 10. For further details see Wolfram MathWorld. In particular, the numerical values of the zeros of the Bessel functions and their first derivatives can be found here.

Displayed below are graphs of the first five Bessel functions of the second kind, Yn(x), over the same range.

For further details see Wolfram MathWorld.


Displayed above are caustics arising from the exponential map on a spheroid x2/4 +y2+z2=1. For example, the primary caustics are drawn in yellow. The mathematical properties of the union of all primary caustics is an open question, whose answer might lie in Sturm-Liouville theory. For more details, see the HCRP project webpage, courtesy of Oliver Knill and Michael Teodorescu.


Temperature profile in a hot plate determined from the solution to Laplace's equation. Details on the boundary conditions can be found here. This webpage was created by Rudi Winter.


Mode of vibration of a circular membrane. The nodes are related to zeros of the Bessel function J3. More details and links can be found here. This webpage was created by Paul Nylander.

For a more interactive demonstration of the vibrational modes, check out this applet which simulates waves in a membrane (like a drum head), showing its various vibrational modes (which you access by clicking on the different squares). Many thanks to John Crewe, who discovered the link to this applet.


Polar plots of some representative spherical harmonics Yℓm(θ,φ), produced by P.Wormer using MATLAB software. This figure is more easily viewed from here. The superscript c (or s) in the figure means that the real (or imaginary) part of the corresponding spherical harmonic has been taken. This notation is used since Re eimφ=cos φ and Im eimφ=sin φ. The overall normalization factors have been omitted as they do not alter the shapes of the polar plots.

There is a very nice demonstration project produced by Stephen Wolfram, the inventor of Mathematica, which allows the user to vary ℓ and m to see how the polar plots of the spherical harmonics change. Check out the Spherical Harmonics demonstration. You can download the relevant CDF file and view it on your computer with the freely available Wolfram CDF Player.


Suppose a large, thin copper plate is at some uniform temperature. Now we imagine a very tiny device that pours out heat: We embed the device at a point in the plate. The resulting plate temperature will spike upward toward infinity at the heat source, and drop off sharply in all directions. This is the Green function for Poisson's equation for heat flow, which is depicted in the figure above. For some interesting details on the life of George Green, check out this website.


This image was taken from the excellent Wikipedia article on the Monty Hall problem. For an entertaining book that describes the history and many variants of this problem, check out the book by Jason Rosenhouse entitled The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser. You can play Let's Make a deal on the web. Click here to see if you win the brand new car!



The top figure exhibits the probability distribution for the sum of the spots on two dice after a roll. The bottom figure presents a histogram of an actual simulation involving 10,000 trials. By comparing the histogram with the predicted probability, one can instantly determine how well the sample distribution matches the predicted probability distribution. Note a typographical error in the actual count corresponding to an outcome of 8. Presumably, it should be something like 1390 and not 1670. For further details, check out the interesting website created by Cary Rhode.

This page contains all class handouts and other items of interest for students of Physics 116C at the University of California, Santa Cruz.

SPECIAL ANNOUNCEMENTS

new!!! Final grades for Physics 116C have been assigned. The final exam grades and the final course grades are listed in this PDF file by student ID in numerical order. The distribution of all the final course grades is shown below:

course grade distribution

Letter grades are based on the cumulative course average, which is weighted according to: homework (9 problem sets with the low score dropped)---40%; midterm exam---25%; and the final exam---35%. This cumulative course average is then converted into a letter grade according to the following approximate numerical ranges: A+ (90--100); A (84--90); A- (78--84); B+ (72--78); B (66--72); B- (60--66); C+ (54--60); C(48--54); D(42--48); F (0--42). Here is the statistical summary of the distribution of the cumulative course averages:

              mean: 64.0               median: 63.5               standard deviation: 13.7               high: 94.5               low : 26.3

Solutions to the final exam (and a link to the final exam statistics) can be found in Section VI on this website.

If you wish to pick up any graded homework or the final exam, please stop by ISB 329 before noon on Wednesday December 14. Angelo Monteux will be available to hand back your graded work. The graded exam will be available for pickup starting on Monday December 12. Homework set 9 will be returned next academic quarter (since the homework grader had to leave town earlier than originally planned). Meanwhile, I will be in England from December 11--21. I will be in my office on Thursday December 22 if you need to consult with me. Otherwise, have a happy and restful holiday season, and I will see you next year!

This website will be kept live for the remainder of the academic year 2011--2012.


Physics 116C: Mathematical Methods in Physics III


I. General Information and Syllabus

The General Information and Syllabus handout is available in either PDF or Postscript format     [PDF | Postscript]
Some of the information in this handout is reproduced below.

Class Hours

Lectures: Tuesdays and Thursdays, 2--3:45 pm, Physical Sciences Building, Room 110

Discussion Section

Mondays, 5--6:10 pm in Earth & Marine Sciences Building, Room B214

Instructor Office Hours

Mondays, 2--4 pm.
In addition, I will be available in my office from 5:00--6:30 pm on most Tuesdays and Thursdays.

Required Textbook

Mathematical Methods in the Physical Sciences, by Mary L. Boas

The author keeps track of the latest list of errata for her textbook in the following PDF file.

Course Grading and Requirements

40% Weekly Homework (9 problem sets)
25% Midterm Exam (Thursday, November 3, 2011, 2--3:45 pm)
35% Final Exam (Thursday, December 8, 2011, 8--11 am)

Homework assignments will be posted on the course website on a weekly basis, and are due at the beginning of class on the due date specified on the assignment sheet. The homework problem sets are not optional. You are encouraged to discuss the class material and homework problems with your classmates and to work in groups, but all submitted problems should represent your own work and understanding. In order that homework can be graded efficiently and returned quickly, there will be a 50% penalty for late homework. This penalty may be waived in special circumstances if you see me before the original due date. Homework solutions will be posted to the course website one or two days after the official due date; no late homeworks will be accepted after that.

There will be one midterm exam and one final exam:

All exams will take place in Physical Sciences Building, Room 110. The midterm exam will be 1 hour and 45 minutes long and cover the first half of the course. The final exam will be three hours long and cover the complete course material. You must take the final exam to pass the course. You will be permitted to consult the class textbook, your own handwritten notes, and any class handout during the exams.

Course Syllabus for Physics 116C

A snapshot of the course lecture schedule appears below. The dates given below are approximate.

Brief Course Outline for Physics 116C

TOPIC Lecture dates Readings
Series solutions of differential equations Sept 22 Boas Chapter 12, sections 1, 11
Legendre polynomials and functions Sept 27, 29, Oct 4 Boas, Chapter 12, sections 2--10
Bessel functions Oct 4, 6, 11 Boas, Chapter 12, sections 12--18, 20
Fuchs' Theorem and the Sturm-Liouville problem Oct 13 Boas, Chapter 12, sections 19, 21
Hermite and Laguerre functions and polynomials Oct 18 Boas, Chapter 12, section 22
Partial differential equations of mathematical physics Oct 20, 25 Boas, Chapter 13, sections 1--4
Problems with cylindrical and spherical symmetry Oct 27, Nov 1 Boas, Chapter 13, sections 5--7
MIDTERM EXAM Nov 3 September/October lectures
Potential theory and Green function techniques Nov 8 Boas Chapter 13, section 8
Integral transform solutions of partial differential equations Nov 10 Boas Chapter 13, section 9
Theory of probability Nov 15, 17 Boas Chapter 15, sections 1--4
Random variables and probability distributions Nov 22, 29 Boas Chapter 15, sections 5--9
Statistics and experimental measurements Dec 1 Boas Chapter 15, section 10
FIANL EXAM Dec 8 entire course material

A summary of the course syllabi for Physics 116A,B,C is available in either PDF or Postscript format     [PDF | Postscript]


Links to the 2011 Physics 116A,B webpages

For your convenience, I am maintaining the Physics 116A webpage established during the winter 2011 academic quarter.

A link to the Physics 116B webpage is also provided courtesy of Stefano Profumo.

Back to the Top

II. Learning Support Services

Learning Support Services (LSS) offers course-specific tutoring that is available to all UCSC students. Students meet in small groups (up to 4 people per group) led by a tutor. Students are eligible for up to one-hour of tutoring per week per course, and may sign-up for tutoring by clicking on this link beginning October 4th at 10:00am. For further information, consult the following FAQ sheet.

The LSS tutor assigned to Physics 116C is Benjamin Michlin (e-mail: bmichlin@ucsc.edu). LSS tutoring sessions will take place on Tuesdays from 4--5 pm. The meeting point is in the S&E library at the entrance to the left, before the stairs. Additional session may be opened based on demand.

Back to the Top

III. Computer algebra systems

Although the use of computer algebra is not mandatory in this class, it can be a very effective tool for pedagogy. In addition, if used correctly, it can be an invaluable problem solving tool. Two of the best computer algebra systems available are Mathematica and Maple.

There are student versions of both Mathematica and Maple available, which sell for the cost of a typical mathematical physics textbook. If you do not wish to invest any money at this time, you can use Mathematica for free at computer labs on campus. For further information click here. A complete list of all the software available in the Learning Technologies computer labs, along with the file path to find the programs on each workstation can be found at this link.

Back to the Top

IV. Homework Problem Sets and Exams

Problem sets and exams are available in either PDF or Postscript format.

Back to the Top

V. Practice Problems for the Midterm and Final Exams and their solutions

Practice midterm and final exams can be found here. These should give you some idea as to the format and level of difficulty of the exam. Solutions will be provided two days before the corresponding exam.

Back to the Top

VI. Solutions to Homework Problem Sets and Exams

The homework set and exam solutions are available in either PDF or postscript format. Solutions will be posted one or two days after the homework is due and after the exams have been completed. Solutions to homework sets are provided by Angelo Monteux, with some further editing by Howard Haber. There is no password required to view the solutions.

Back to the Top

VII. Other Class Handouts

Back to the Top

1. The Wronskian is especially useful in the theory of ordinary linear differential equations. If y1(x), y2(x),..., yn(x) are linearly independent solutions to the differential equation, then their Wronskian is nonvanishing. Even if the solutions cannot be determined analytically, one can determine the Wronskian using Abel's formula. This formula is derived in this handout entitled Applications of the Wronskian to ordinary linear differential equations, and a number of other applications of the Wronskian are exhibited.   [PDF | Postscript].

2. A handout entitled Series solutions to a second order linear differential equation with regular singular points provides details on how to evaluate the corresponding series and determine the two linearly independent solutions. In particular, if the roots of the indicial equation are equal, then one of the linearly independent solutions includes a logarithmic term. The logarithmic case can also arise if the roots of the indicial equation differ by an integer. After providing the general formulae for the various cases, examples are given using the Bessel differential equation to illustrate all the possibilities.   [PDF | Postscript].

3. A handout entitled Spherical Harmonics examines the solutions to Laplace's equations in spherical coordinates. These solutions possess a radial and angular part; the latter are called the spherical harmonics, Yℓm(θ,φ), where ℓ=0,1,2,3,... and m=-ℓ,-ℓ+1,...,ℓ-1, ℓ. These notes provide a discussion of the properties of the spherical harmonics, along with the famous addition theorem. Two proofs of the addition theorem are given (one in an appendix), and the application to the expansion of the inverse distance function is provided.   [PDF | Postscript].

4. A handout entitled The Laplacian of the inverse distance provides a derivation of ∇2(1/r)=-4πδ3(r). This result is then applied to solving Poisson's equation. These notes also introduce the concept of the inverse Laplacian and discuss its relation to the Green function, along with some interesting applications.   [PDF | Postscript].

5. A collection of independent and identically distributed (or iid for short) random variables are a set of random variables with the same probability distribution (or density). These are suitable for modeling independent trials. If x1, x2,..., xn are n iid random variables, then E(xi)=μ and Var(xi)=σ2, for any value of i=1,2,3,...,n. The aveage of the n random variables is defined by x=(x1+ x2+...+xn)/n. In class, I derived the results E(x)=μ and Var(x) =σ2/n. Details of the derivation are conveniently summarized in the following note by C. Alan Bester from the University of Chicago.   [PDF]   This note refers to slides that can be found here, although you should be able to confirm each step based on the results obtained in class.

6. The standard deviation of the mean (also called the standard error) provides an estimate of the uncertainty of the experimental determination of the expected value of a random variable. In the case of n measurements, standard error is a factor of n smaller than the standard deviation of the random variable. The latter corresponds to the uncertainty in a single measurement. This distinction is clarified in a class handout entitled The Standard Deviation of the Mean.     [PDF | Postscript]

VIII. Free online textbooks

1. A very good source for free mathematical textbooks can be found on FreeScience webpage. I can also provide another useful link to a list of free mathematics textbooks.

2. One of the most useful undergraduate textbooks on mathematical physics is: Essential Mathematical Methods for Physicists, by Hans J. Weber and George B. Arfken, which is available for online viewing here. It takes a while to load, so be patient.

3. Sean Mauch (of Caltech) provides a free massive textbook (2321 pages) entitled Introduction to Methods of Applied Mathematics. However, many of the topics of Physics 116C, if treated by Mauch, are in a preliminary status.

4. James Nearing also provides a free textbook entitled Mathematical Tools for Physics. This book treats some of the topics of Physics 116C at a similar level of difficulty.

5. Russel L. Herman has provided notes for a second course in differential equations, covering linear and nonlinear systems and boundary value problems. His monograph is entitled A Second Course in Ordinary Differential Equations: Dynamical Systems and Boundary Value Problems. Of particular interest to Physics 116C are Chapter 6 on Sturm Liouville problems, Chapter 7 on special functions (including Bessel functions and a variety of orthogonal polynomials) and Chapter 8 on Green functions.

6. A textbook entitled Partial Differential Equations for Scientists and Engineers, by Geoffrey Stephenson is available from the following PDF link.

7. A textbook entitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell has been generously made available free of charge from the following PDF link, along with PDF solutions to all odd-numbered problems. I especially recommend Chapter 4 for its superb treatment of conditional probability, with applications to the Monty Hall problem, the two-child paradox, and other purported paradoxes that arise in probability theory.

Back to the Top

IX. Articles and Books of Interest

Back to the Top

1. All the examples of the indicial equations given in Boas possess real roots. But a quadratic equation with real coefficients can also have complex roots. Applying the Frobenius method in the case of complex indicial roots is treated pedagogically in a paper entitled The Frobenius method for complex roots of the indicial equation by Joseph L. Neuringer, International Journal of Mathematical Education in Science and Technology, Volume 9, Issue 1, 1978, pages 71--77.     [PDF]

2. If you were wondering whether the Laguerre differential equation can also be solved by factorizing the equation into a product of first order differential operators and constructing ladder (raising and lowering) operators, check out the article by H. Fakhri and A. Chenaghlou entitled Ladder operators for the associated Laguerre functions in J. Phys. A: Math. Gen. 37 (2004) pp. 7499--7507.     [PDF]

3. The factorization method for solving differential equations was made popular by physicists. The method was described in a classic review paper by L. Infeld and T.E. Hull entitled The Factorization Method in Reviews of Modern Physics 23 (1951) pp. 21--68.     [PDF]

4. One of the best elementary treatments of partial differential equations suitable for physicists can be found in a book by Gabriel Barton, Elements of Green's Functions and Propagation---Potentials, Diffusion and Waves, published by Oxford Science Publications in 1989. Among other things, there is a very clear discussion of which boundary conditions constitute a well-posed problem.

5. Spherical harmonics play a critical role in computer graphics. Colorful visual representations of the spherical harmonics provide added insight into their properties and significance. See for example, the following two articles:

6. In the Monty Hall problem, there are three doors. Behind one of them is a brand new car, whereas the other two doors conceal goats. Monty Hall asks you to choose a door without opening it. Then, he opens one of the other doors to reveal a goat. At this point, Monty asks you whether you wish to switch your choice of doors or to stand pat. Indeed, what is the probabilty of winning the car if you switch your choice of doors? When Marylin vos Savant answered this question in the September 9, 1990 issue of Parade magazine, the reader response was overwhelming, many of whom could not believe her proposed answer that switching resulted in a probability of 2/3 for winning the car. Wikipedia provides an excellent introduction to this problem. For a more detailed account, read about the history of the Monty Hall problem and its many variants in this enertaining book by Jason Rosenhouse entitled The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser (Oxford University Press, Oxford, UK, 2009).

7. Apparent paradoxes often arise in probability theory. One such paradox is the called the two-child paradox, which asks for the probability that a two-child family has two boys given that one of the children is known to be a boy. Once again, Wikipedia provides an excellent introduction to this problem. In addition, check out a fascinating article by Peter Lynch, The Two-Child Paradox: Dichotomy and Ambiguity in Irish Mathematical Society Bulletin 67 (2011) pp. 67--73     [PDF].

8. The Monty Hall problem and the two-child paradox can be analyzed by using conditional probabilites. I highly recommend Chapter 4 of the free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell previously cited in Section XIII of this website. Another excellent treatment of these and other well-known puzzles in probability can be found in an article by Maya Bar-Hillel and Ruma Falk, Some teasers concerning conditional probabilities in Cognition 11 (1982) pp. 109--122     [ PDF].

9. The law of large numbers and the central limit theorem are treated at an elementary level in Chapters 8 and 9, respectively of Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell, previously cited in Section XIII of this website.

10. David L. Streiner provides an article entitled Maintaining Standards: Differences between the Standard Deviation and Standard Error, and When to Use Each, which should help illuminate the distinction between the standard deviation and the standard error (also known as the standard deviation of the mean).     [PDF]

X. Other Web Pages of Interest

1. A superb resource for both the elementary functions and the special functions of mathematical physics is the Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun, which is freely available on-line. The home page for this resource can be found here. There, you will find links to a frames interface of the book. Another scan of the book can be found here. A third independent link to the book can be found here.

2. The NIST Handbook of Mathematical Functions (published by Cambridge University Press), together with its Web counterpart, the NIST Digital Library of Mathematical Functions (DLMF), is the culmination of a project that was conceived in 1996 at the National Institute of Standards and Technology (NIST). The project had two equally important goals: to develop an authoritative replacement for the highly successful Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards (M. Abramowitz and I. A. Stegun, editors); and to disseminate essentially the same information from a public Web site operated by NIST. The new Handbook and DLMF are the work of many hands: editors, associate editors, authors, validators, and numerous technical experts. The NIST Handbook covers the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas.

3. Another very useful reference for both the elementary functions and the special functions of mathematical physics is An Atlas of Functions (2nd edition) by Keith B. Oldham, Jan Myland and Jerome Spanier, published by Springer Science in 2009. This resource is freely available on-line to students at the University of California at this link.

4. Yet another excellent website for both the elementary functions and the special functions of mathematical physics is the Wolfram Functions site. This site was created with Mathematica and is developed and maintained by Wolfram Research with partial support from the National Science Foundation.

5. One of the classic references to special functions is a three volume set entitled Higher Transcendental Functions (edited by A. Erdelyi), which was compiled in 1953 and is based in part on notes left by Harry Bateman. This was the primary reference for a generation of physicists and applied mathematicians, which is colloquially referred to as the Bateman Manuscripts. Although much of the material goes considerably beyond the level of Physics 116C, this esteemed reference work continues to be a valuable resources for students and professionals. PDF versions of the three volumes are now available free of charge. Check out the three volumes by clicking on the relevant links here:     [Volume 1 | Volume 2 | Volume 3].

6. Google is invaluable for searching for mathematical information. For example, if I need the first few Bessel function zeros, I search with google for "Bessel function zeros." The first hit is Bessel function zeros on the Wolfram MathWorld site.

7. Wikipedia provides some good articles on various mathematical topics. For example, a simple exposition on the the law of large numbers can be found here. The Wikipedia treatment of central limit theorem is more technical. What I discussed in class was the classical CLT (or Lindeberg-Levy CLT), which is presented in the first paragraph of the Wikipedia article. However, Wikipedia has another page that illustrates the central limit theorem which is quite illuminating. Check it out at this link.

Back to the Top

haber@scipp.ucsc.edu
Last Updated: December 11, 2011