math121b:start

# Math 121B: Mathematics for Physical Sciences

UC Berkeley, Spring 2020

Zoom Meeting: https://berkeley.zoom.us/j/486973340 MWF 9-10AM

Zoom Office hours: MWF: 10-11am, and Monday Friday afternoon 4-5pm.

• My Personal Meeting ID is: 881-910-2324.
• Please also join the chat channel, named “Math 121B”. So that if I am not in my 'office' during office hour, you can send me a message, and I will be back.

## Instructor

Peng Zhou (pzhou.math@berkeley.edu)
Office: 931 Evans
Office Hour: MWF: 10-11am, M:12-1pm. W:2-4pm.

## Syllabus

We continue to use the textbook by Boas, Mathematical Methods in the Physical Sciences 3rd edition.

We will cover chapters 10, 11, 12, 13, 15. We split them into two parts

• Part I: Ch 10, 13.1-4. Chapter 10 is about tensor notation and curvilinear coordinates, which will be used in Ch 13 to do separation of variable for Laplace operator $\Delta$, and reduce PDE to ODEs.
• Part II: Chapter 12 is about solving these ODEs, and the solutions are Bessel function and Legendre function. Finally, after these hard works, we can tackle Chapter 13 for various PDEs in physics.
• Part III: Ch 15, We will learn basic probability concepts. If time permits, we will do some topics on probability, such as central limit theorems, stochastic processes, or markov chain.

Exams We will have two midterms and one final. midterms are for part I and II, and final is accumulative.

Homeworks Homeworks will be assigned weekly, but not collected or graded. You are welcome to submit for comments.

Grading Total grade = 30% + 30% for the two midterms + 40% final.

Accomodation If you are a DSP student and need accomodation for exams, please let me know at the beginning of the semester.

## References

These are excerpts of the book Finite Dimensional Vector Spaces, written by Paul R. Halmos.

The whole book can be found in our library with online access, or directly here .

“A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.” – Paul Halmos

## Problems

• Ex 5: Legendre functions in Ch 12. Problems
• 1.7, 1.10, 4.4, 5.1, 5.4, 5.5, 5.8-10,
• 6.6, 7.1, 7.2, 7.6, 8.2,
• 9.1(Hint: use Rodrigue formula and integration by part), 9.3
• 10.3, 10.5, Problem 10.7
• Ex 6: Bessel Function in Chapter 12
• 12.1, 12.4, 12.6. Using series expansion of Bessel function to show certain recursion relation.
• 13.3. This is about 'special Bessel function' later.
• 15.6, 15.7, 15.8. Application of the recursion relations.
• 16.2, 16.4, 16.5, 16.16
• 17.6, 17.7
• 19.1 (derive the orthogonality and normalization equation)
• 20.1-6, these exercises allows one to be familiar with asymptotic behavior of these functions
• 21.1, 2, 5
• section 22, read the text.
• Ex 7: Ch 13
• 1.3, 2.1, 2.3, 3.2, 3.4, 4.2, 4.5, 5.2, 5.8

## Lectures

• Week 1
• 2020-01-22, Wednesday: What is a vector? We will review 3.10 and 3.14. A vector space without choosing a basis is quite OK.
• 2020-01-24, Friday: What is a tensor? The concrete way (10.2, 10.3) and the abstract way.
• Week 2
• Week 3
• Week 4
• Week 5
• Week 6
• Week 7
• 03-02: More on generating function. Orthogonality of Legendre polynomial. Boas 10.5 - 10.7
• 03-04: Associated Legendre function. Boas 0
• 03-06: Started Bessel functions.
• Week 8:
• Week 9:
• Week 11
• Week 12
• 04-10: ipad note (corrected a mistake about orthogonality of $P_l^m(x)$.
• Midterm 2: take home midterm, from Friday evening to Sunday midnight. Exam will be release through piazza. The test will cover Boas chapter 11 (Beta function and Gamma function), 12, 13 (except Integral transform section). It will not cover things outside of Boas, example: steepest descend method.
• Week 13
• For the remaining 3 week, we will study probability.
• This week, we will cover basic concepts. Here is a summarizing note.
• SageMath / CoCalc: SageMath is an open source math programming platform, contains python and R. CoCalc is an online server supporting it. So you don't have to install anything on your computer.
• Programming with R: read page 1-7 to get started. intro
• Week 14: Famous continuous and discrete distribution
• Week 15
• 04-27: note introduction to stochastic process, following the book by Dobrow. Available online in library. The code used in today's lecture is from here
• 04-29: note Law of Large number, Central Limit Thm. Begin Hypothesis testing.
• 05-01: note