Lecture Notes

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======  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. 

** Piazza ** Please sign up at https://piazza.com/berkeley/spring2020/math121b 



===== References =====
These are excerpts of the book //Finite Dimensional Vector Spaces//, written by Paul R. Halmos.  
  * {{ :math121b:dual_space.pdf | Dual vector space }}, 
  * {{ :math121b:direct_sum_bilinear_form_and_tensor_product.pdf | Tensor product, Direct Sum }} 
The whole book can be found in our library with online access, or directly {{ :math121b:halmos.pdf |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 1: [[about vectors and tensors]]
  * Ex 2: [[ex2 | Read the attached note (below in the reference section) on Dual vector spaces, and do exercise on page 22. ]]
  * Ex 3: [[ex3 | Exercises on Tangent Vector and Metric Tensor ]]
  * Ex 4: [[ex4 | curvilinear coordinates]]
  * [[math121b:sample-m1| Sample Midterm 1]]
  * 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, [[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
    * [[01-22]]: What is a vector? We will review 3.10 and 3.14. A vector space without choosing a basis is quite OK. 
    * [[01-24]]: What is a tensor? The concrete way (10.2, 10.3) and the abstract way. 
  * Week 2
    * [[01-27]]: Dual vector space. Inner product and metric tensor. 
    * [[01-29]]: Wedge product. Kronecker $\delta_{ij}$ and Levi-Cevita Symbol $\epsilon_{ijk}$. 
    * [[01-31]]: Review of concepts and did one example
  * Week 3
    * [[02-03]]: Curvilinear coordinates. Tangent Vector,
    * [[02-05]]: Metric Tensor in Curvilinear Coordinate. 
    * [[02-07]]: Volume, Cotangent Vector. Laplacian
  * Week 4
    * [[02-10]]: finish loose ends. Start Boas. 
    * 02-12: will finish Monday's lecture note. 
    * [[02-14]]: Begin Ch 13. 
  * Week 5
    * [[02-19]]: Separation of Variable. 
    * [[02-21]]: 
  * Week 6
    * Midterm 1: Chapter 10. You may bring notes to exam.  {{ :math121b:math_121b_midterm1_sol.pdf | Midterm 1 and Solution}}
    * [[02-26]]: Start Chapter 
    * 02-28: Review Exam problems. Rodrigue formula. Generating functions. Read: Boas , 12,5
  * 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: 
    * 03-09: Discussed the Exercises on Legendre polynomial 
    * [[03-11]]: Bessel functions. 
      * Zoom meeting. [[https://berkeley.zoom.us/j/569471250]] 
    * 03-13: {{ :math121b:math_121b_note_13_mar_2020.pdf | note}} Boas section 6, 9. 
  * Week 9: 
    * 03-16: {{ :math121b:note_16_mar_2020_121b.pdf | note}} Beta function 1 11.7
      * Optional: derivation of [[https://www.nbi.dk/~polesen/borel/node14.html#hankel | Hankel representation of Gamma function]], and [[https://www.nbi.dk/~polesen/borel/node15.html | Bessel function]] (Eq 144, 145, 146).  
    * 03-18: {{ :math121b:note_18_mar_2020.pdf | note}} Generating Function of Bessel function. Begin Stationary phase expansion. 
      * HW exercises: derive the residue formula $\oint_{|u|=1} u^{-1} du = 2\pi i$. Try compute $\int_{-\infty}^\infty e^{ - e^{i \theta} x^2/2} dx$ for $|\theta| < \pi/2$. 
      * An online note about [[https://www.encyclopediaofmath.org/index.php?title=Saddle_point_method | stationary phase method]] (aka, method of steepest descend, saddle point method). 
    * [[03-20]]: Steepest Descent Method for obtaining the asymptotic behavior of the Bessel function. {{ :math121b:note_20_mar_2020.pdf | note during class}}
  * Week 11
    * [[03-30]], {{ :math121b:note_30_mar_2020.pdf |ipad note.}} 
    * [[04-01]], {{ :math121b:note_1_apr_2020.pdf |ipad note}}
    * [[04-03]]
  * Week 12
    * [[04-06]]
    * 04-08: {{ :math121b:note_8_apr_2020.pdf |ipad note}}
    * 04-10: {{ :math121b:note_10_apr_2020.pdf |ipad note}} (corrected a mistake about orthogonality of $P_l^m(x)$. 
    * [[midterm2 | 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. 
    * [[midterm2-solution]]
  * Week 13
    * For the remaining 3 week, we will study probability. 
    * This week, we will cover basic concepts. Here is a [[http://ai.stanford.edu/~paskin/gm-short-course/lec1.pdf | summarizing note]].
    * [[04-15]]{{ :math121b:note_15_apr_2020.pdf |ipad note}}. Read Boas 15.1 - 15.5
    * SageMath / CoCalc: [[https://www.sagemath.org| SageMath]] is an open source math programming platform, contains python and R. [[https://cocalc.com/| 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. [[https://cran.r-project.org/doc/contrib/Verzani-SimpleR.pdf|intro]] 
  * Week 14: Famous continuous and discrete distribution
    * 04-20: [[R]], {{ :math121b:note_20_apr_2020.pdf |ipad note}}, computer examples{{ :math121b:welcome_to_cocalc.pdf |}}. 
    * 04-22: {{ :math121b:note_22_apr_2020.pdf |ipad note}}. Suggested problems. Boas 15.11 #1 - #4, #10, #11. 
    * 04-24: Inequalities. {{ :math121b:note_24_apr_2020.pdf | note}}. And some R exercise, following [[https://cran.r-project.org/doc/contrib/Verzani-SimpleR.pdf|appendix]] 
  * Week 15
    * 04-27: {{ :math121b:note_27_apr_2020.pdf |note}} introduction to stochastic process, following the  [[https://www.amazon.com/Introduction-Stochastic-Processes-Robert-Dobrow/dp/1118740653 |book ]] by Dobrow. Available online in library. The code used in today's lecture is from [[https://people.carleton.edu/~rdobrow/stochbook/RScripts.html | here]]
    * 04-29: {{ :math121b:note_29_apr_2020.pdf |note }} Law of Large number, Central Limit Thm. Begin Hypothesis testing. 
    * 05-01: {{ :math121b:note_1_may_2020.pdf |note }}
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