https://courses.pzhou.org/ 2026-04-17T10:03:05+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:august_23&rev=1771684874&do=diff Aug 23: what is math for physics and engineering? Is math useful for physics? Well, math is born out of physics. As the Russian mathematical physicist V.I. Arnold says. Is math useful for chemistry? I don't know much about chemistry, except that electron rotates around nuclei, and one needs some wavefunction to describe the quantum states of electron. So chemistry needs physics, hence needs math? text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:august_25&rev=1771684874&do=diff Review of Linear Algebra Let's start from scratch again. What is linear algebra? This is a textbook on linear algebra by Prof Givental. What is a vector space Answer 1 row vectors, column vectors, matrices. Let's also review the index notation $a_i = \sum_{j} M_{ij} b_j$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:august_28&rev=1771684874&do=diff August 28: Review Linear algebra 2 Today we will continue our review of linear algebra. Hopefully you have brushed up on the set notations over the weekends. quotient space Let $V$ be a vector space, and $W \In V$ be a subspace. The quotient space $V/W$ is the following vector space: text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:august_30&rev=1771684875&do=diff August 30: Review of Calculus today we will go over sequence of numbers and limit. sequence Let $a_1, a_2, \cdots $ be a sequence of numbers. We can have many examples of it. * $1,-1,1,-1\cdots $ * $0.9,0.99, 0.999, $ * $1,2,3, \cdots, $ limit text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:blog&rev=1771684874&do=diff text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:final-distribution&rev=1771684875&do=diff 89 85 82 81 70 67 65 63 63 60 57 51.5 51 50 49.5 45 45 43 42 39 38 34 32.5 30.5 29 28 24 23 21 20 19 18 15 6 text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_1&rev=1771684875&do=diff Homework 1 For all the following exercises, if you feel it is too easy, skip it; if you find it interesting and relevant, do it; if you find it too hard, ask about it on discord, let's tackle it together. 1. If you are not familiar with set theory notation and terminology, watch these two short videos: text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_2&rev=1771684874&do=diff Homework 2 [ solution] (thanks to an anonymous student who provided the solution) I will update the homework after each lectures. It is due next Wednesday (since we have Labor day Monday) Vector Space Problems 1. Let $V \In \R^3$ be the points that $\{(x_1, x_2, x_3) \mid x_1 + x_2 + x_3=0\}$. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis. text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_3&rev=1771684875&do=diff Homework 3 (Due Wednesday, Sep 13) 0. Read Boas Ch2, section 1 - 9, find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing). 1. let $z = 2 e^{i \pi / 3}$, * compute $z^2, z^3$. text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_4&rev=1771684875&do=diff Homework 4 Due Monday in class. 1. Taylor expand $(z+1)(z+2)$ around $z=3$. 2. Laurent expand $1/[(z-1)(z-2)]$ around $z=1$. And do it again, this time around $z=2$. 3.Compute $\int_{0}^{2\pi} 1 / (z(t)) d z(t). $ for the following three contours (a) For $t \in [0, 2\pi]$, let $z(t) = e^{it}$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_5&rev=1771684874&do=diff Homework 5 Due Monday in class 1. Let $C$ be the contour of $|z|=10$. Compute the following integrals. (1) $$\oint_C \frac{1}{1+z^2} dz $$ (2) (the result for this one is not zero.) $$\oint_C \frac{z}{1+z^2} dz $$ (3) $$\oint_C \frac{z^2}{1+z^4} dz $$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_6&rev=1771684874&do=diff Homework 6 Due on Monday (Oct 9th) 1. Find the Fourier transformation of the following function. $$ f(x) = \begin{cases} 1 & 0<x<1 \cr 0 & \text{else} \end{cases} $$ 2. Find the Fourier transformation of the following function. $$ f(x) = \begin{cases} 1+x & -1<x<0 \cr 1-x & 0<x<1 \cr 0 & \text{else} \end{cases} $$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_7&rev=1771684874&do=diff Homework 7 (Due next Wednesday) We will use the Boas convention for Fourier transformation (or see Friday's note). 1. Discrete Fourier Transformation for $N=3$. Suppose $f(x)$ is given by $$ f(x) = \delta_{x,0} $$ where $\delta_{i,j} = 0$ is $i\neq j$ and $=1$ if $i=j$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_8&rev=1771684874&do=diff Homework 8 1. Sine and Cosine decomposition. Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$. Such function $f(x)$ can be expressed as a sum of 'sine waves' and cosine waves and constant $$ f(x) = a_0 + \sum_{n=1}^\infty a_n \cos(2n \pi x) + b_n \sin(2n \pi x). $$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:hw_9&rev=1771684874&do=diff HW 9 Boas * section 8.5, #22, #24, #27 * section 8.8 #8, 9, 10 * section 8.9 #2, 3 text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:midterm_1&rev=1771684874&do=diff Midterm 1 Range This is in class, 50 minutes midterm. We will have 5 problems, * 1 on sequence and series * 1 on basic complex number geometry * 1 on Taylor expansion and Laurent expansion. * 2 on complex contour integral. Here is a past midterm 1 (in 2019. That exam was 80 minutes, longer than what we will have) text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:mt1-distribution&rev=1771684875&do=diff 49 48 45.5 44.5 44 42 39.5 37.5 37.5 36.5 35 33 33 32 31.5 31.5 29 29 28.5 28.5 26 25.5 24.5 20.5 19.5 18.5 18.5 18 17.5 17 15.5 15 10 8.5 7.5 text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:mt2-distribution&rev=1771684874&do=diff 50 50 50 50 49.5 48 47.5 47 47 47 46.5 46.5 46.5 46.5 45 43.5 43 42.5 42.5 42.5 41 38 38 37.5 37 37 37 36.5 34 32 29 19 13 12 10 7 text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:november_1_wednesday&rev=1771684874&do=diff Nov 1st, other types of ODE we covered the idea of Green's function in more detail. The other types of ODE will be dealt with after midterm 2. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:november_3_friday&rev=1771684874&do=diff Review for Midterm 2 We will go over part of the past exam [ 2019 fall midterm 2], [ solution] We will have some more Fourier transformation problem (as in the homework problems), not so much contour integral there (which was covered in midterm 1). text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_4_wednesday&rev=1771684875&do=diff October 4 (Wednesday) What's coming in the second part of this course? * Fourier transform. Given $f(x)$, we want to express $f(x) = \int e^{ikx} g(k) dk$, since $e^{ikx}$ is easy to deal with. * Laplace transform. Given $f(t)$, on $t \in [0, \infty)$, we want to write $f(t) = \int_{c + i \R} e^{pt} F(p) dp$. text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_6_friday&rev=1771684875&do=diff October 6 (Friday) Topics: * Fourier inversion formula * Fourier Series for periodic function * Interpretation of complex vector space, hermitian inner product, orthonormal basis. Inversion Formula Suppose you started from $f(x)$, and did some hard work to get the Fourier transformation $F(p)$. Can you recover $f(x)$ from $F(p)$? Did you lose information when you throw away $f(x)$ and only keep $F(p)$? text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_11_wednesday&rev=1771684874&do=diff October 11 (Wednesday) We will do many examples today to get intuition for what's Fourier transform is doing. Discrete Fourier Transform General Formula Let $N$ be a positive integer. Let 'x-space' be $V_x = \Z/ N \Z = \{0,1, \cdots, N-1\}$, and then (rescaled) 'p-space' is also $V_p = \Z / N\Z$, then we can build the Fourier transformation kernel $$ K(x,p) = e^{2\pi i (x p / N)} : V_x \times V_p \to U(1) $$ where $U(1)$ is the unit circle in complex number. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_13_friday&rev=1771684874&do=diff October 13, Friday Parseval Equality says, Fourier transformation, as a linear map from one function space (function on x), to another function space (function on p), preserves 'norm'. Norm is just a fancy way of saying 'length of a vector'. What do we mean by the length of a function? text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_16_monday&rev=1771684874&do=diff October 16 (Monday) * What is Laplace transform? * What's the difference between that and Fourier transform? * When to use it? Definition Given a function $f(t)$ on the positive real line $t>0$, we can define the following function of $p$: $$ F(p) = \int_{t=0}^\infty f(t) e^{-pt} dt. $$ Again, we require the function $f(t)$ to have moderate growth at $t \to \infty$ for the integral to be well-defined. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_20_friday&rev=1771684874&do=diff Oct 20. Exercise Day! So far, we have learned many transformations and inverse transformation. Conceptually, we are just trying to decompose a given function $f(x)$ as a linear combination of $e^{ax}$ for various $a$. Because it is an eigenfunction of $(d/dx)$: $$ (d/dx) e^{ax} = a e^{ax}$$ text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_23_monday&rev=1771684875&do=diff Oct 23: constant coeff diffrential equation warm-up: how to solve the equation $$ (z-a) (z-b) = 0 $$ well, we would have two solutions $z=a$ and $z=b$ (assuming $a \neq b$). 1 how to solve equation ( $a \neq b$) $$ (d/dx - a) (d/dx - b) f(x) = 0? $$ Instead of saying we have two solutions, we say the solution space is a two dimensional vector space. it is easy to check that $e^{ax}$ and $e^{bx}$ solves the equations, and so are their linear combinations, thus we can write down the genera… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_25_wednesday&rev=1771684874&do=diff Oct 25: Wednesday Today we considered solving homogenous constant coefficient differential equation. In general, the equation you meet looks like $$ (d/dx)^n f(x) + c_{n-1} (d/dx)^{n-1} f(x) + \cdots + c_1 (d/dx) f(x) + c_0 f(x) = 0. $$ If we call $D = d/dx$ and factor out $f(x)$ on the right, we can write the above equation as $$ (D^n + c_{n-1} D^{n-1} + \cdots + c_0 ) f(x) = 0. $$ We sometimes use $P(D) = D^n + c_{n-1} D^{n-1} + \cdots + c_0$, $P(D)$ is a degree $n$ polynomial in $D$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_27_friday&rev=1771684874&do=diff Oct 27 We learnd Laplace transformation, which can be used to solve diff eq. Let $f(t)$ be a function defined for $t>0$, and we recall the following $$ F(p) = [LT(f)] (p) = \int_0^\infty f(t) e^{-pt} dt. $$ and the inverse Laplace transformation is $$ f(t) = (1/2\pi i) \int_{c-i \infty}^{c+i \infty} F(p) e^{pt} dp. $$ for $c \gg 1$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:october_30_monday&rev=1771684874&do=diff Oct 30 Sometimes you want to model a short impulse: * you give a swing a push, so that it started to swing up * you plucked a string on the guitar, and it started to vibrate This can be modeled by an inhomogenous equation of the form $$ D^2 u(t) + \omega^2 u(t) = g(t). $$ where $g(t)$ models the force (which varies over time). Sometimes we don't care about the precise shape of the function $g(t)$, but only the 'total effect' of the force given, then it is useful to use the delta function… text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_1&rev=1771684875&do=diff September 1: Differentiation and Integration Differentiation What is differentiation? it is measuring the ratio of how the output change versus how the input changes. It is a linear map from the vector space of small change of input, to the vector space of small changes of output. text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_6_wednesday&rev=1771684875&do=diff September 6: Introduction to complex numbers 1. what is complex number? (a pair of real numbers, put together, that you can multiply together) 2. why we need it? (I will leave that to you) 3. how to think about it? (Cartesian coordinate, polar coordinate. ) text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_8_friday&rev=1771684874&do=diff September 8th: basic complex functions Today we are going to meet with some old friends, which will serve as anchor when we go out and meet with more exotic ones. 1. exponential and log 2. sin, cos, sinh, cosh (not a big deal, they are linear combination of exp,log) text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_11_monday&rev=1771684874&do=diff Holomorphic Function, Meromorphic Function tl;dr A function $f: \C \to \C \cup \{\infty\}$ is holomorphic at $z_0$ if there exists $\epsilon>0$, and complex numbers $a_0,a_1,\cdots$ that for all $|z - z_0|<\epsilon$, we can write $$ f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n. $$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_13_wednesday&rev=1771684874&do=diff Contour Integral Reading: Boas, Ch14, section 1-5 So, you have learned what holomorphic function looks like, and you know there are functions which are 'bad' only at a few points. What do you want to do with these functions? primitive of a holomorphic function on $\C$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_15_friday&rev=1771684874&do=diff Contour Integral: exercise day Let's see some cool application of contour integral first. 1. $\int_\R 1/(1+x^2) dx$ ---------- didn't finish, the rest will wait till next week. 2. $\int_0^\infty 1/(1+x^3) dx$ 3. $\int_0^\infty e^{-x}/(1+x^3) dx$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_18_monday&rev=1771684874&do=diff September 18: Residue theorem Let $\Omega$ be a domain (open subset), and $z_0 \in \Omega$ a point. We say a holomorphic function $f: \Omega \RM \{z_0\} \to \C$ has a pole of order $m$ at $z_0$, if $h(z) = (z-z_0)^m f(z)$ can be extended to the entire $\Omega$, and $h(z_0) \neq 0$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_20_wednesday&rev=1771684874&do=diff September 20 (Wednesday) Today we discussed this integral $$ \oint_{|z|=10} \frac{1}{(z-1)(z-2)} dz $$ where the contour is a CCW (counter-clockwise) circle of radius $10$ (or any radius $R > 2$, that encloses $1,2$) We used three methods to show that this is zero. Denote the integrand by $f(z)$, namely $f(z) = \frac{1}{(z-1)(z-2)}$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_22_friday&rev=1771684874&do=diff September 22 (Friday) How to think about the 'function' $f(z) = \sqrt{z}$? How to think about $\sqrt{(z-1)(z-2)}$? We will make sense of the graph of these multivalued functions. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_25_monday&rev=1771684874&do=diff September 25 Monday We talked about two integrals, one is $$ \int_{\theta=0}^{2\pi} \frac{1}{1 + \epsilon \cos(\theta)} d\theta, 0 < \epsilon \ll 1 $$ the other is $$ \int_{x=0}^\infty \frac{1}{1+x^n} dx $$ integration of trig function Suppose we have a ration function involving $\sin(\theta)$ and $\cos(\theta)$, $R(\sin \theta, \cos \theta)$, and we consider integral of the form $$ \int_{\theta=0}^{2\pi} R(\sin \theta, \cos \theta) d \theta$$ Then, we can replaced $e^{i\theta} = z$, let $z… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:september_27_wednesday&rev=1771684874&do=diff September 27, Wednesday Gaussian integral is of the form (modulo constant) $$I = \int_{-\infty}^\infty e^{- x^2} dx $$ How to evaluate this? here residue theorem cannot help you. you need to use a little trick. Consider the double integral $$I^2 = \int_{-\infty}^\infty\int_{-\infty}^\infty e^{- x^2 -y^2 } dx dy $$ then switch to radial coordinate $$ I^2 = \int_{r=0}^{\infty}\int_{\theta=0}^{2\pi} e^{-r^2} r drd\theta = 2 \pi \int_{r=0}^{\infty} e^{-r^2} r dr$$ subsitatue $r^2=u$, we get $2r… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121a-f23:start&rev=1771684874&do=diff Math for physics Sciences Berkeley Math 121A. Lecture: MoWeFr 10:00AM - 10:59AM Dwinelle 109 Office Hour: Wednesday 11am-12:30noon, Friday 11am-12:30noon. Evans 753 Final Exam: Dec 11, 8-11am, classroom Dwinelle 109 online discussion : Discord anonymous feedback