https://courses.pzhou.org/ 2026-04-17T10:18:52+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:01-22&rev=1771684876&do=diff 2020-01-22, Wednesday Hi, this is my post-lecture note. A short summary of what we did in class today. (Also a chance to correct my mistakes made in class) So, what is a vector? There are many possible correct answers (in different sense). You may say one of the following text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:01-24&rev=1771684876&do=diff 2020-01-24, Friday What is a tensor? If you say a vector is a list of numbers, $v=(v^1, \cdots, v^n)$, index by $i=1, \cdots, n$, then a (rank 2) tensor is like a square matrix, $T^{ij}$, where $i, j = 1, \cdots, n$. But there is something more to it. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:01-27&rev=1771684876&do=diff 2020-01-27, Monday Last time, we have reviewed the abstract definition of vector space. And we defined the tensor products of any two vector spaces $V$ and $W$, as a new vector space $V \otimes W$. Today, we introduce the notion of the dual vector space text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:01-29&rev=1771684876&do=diff 2020-01-29, Wednesday $$\gdef\ot\otimes$$ Tensor power of a vector space Let $V$ be a finite dimensional vector space. We denote the $k$-th tensor power of $V$ as $$ V^{\otimes k} = \underbrace{V\ot \cdots \ot V}_{\text{ $k$ times} } $$ Its elements are linear combinations of terms like $v_1 \otimes \cdots \ot v_k$, subject to the usual linearity relations. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:01-31&rev=1771684876&do=diff 2020-01-31, Friday Today, we first reviewed what our plan was. We try to answer a few questions they are Where are we going ? Our final goal for part I (material for midterm I) is to derive the Laplace operator in curvilinear coordinate, which you can find the formula text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-03&rev=1771684876&do=diff 2020-02-03, Monday Curvilinear coordinate Definition(Curviliear coordinates) Let $U$ be an open set in $\R^n$. A curvilinear coordinate on $U$ is a smooth function $f=(f_1,\cdots,f_n): U \to \R^n$, such that * $f$ is a bijection between $U$ and its image $f(U)$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-05&rev=1771684876&do=diff $$\gdef\d\partial$$ 2020-02-05 Metric Tensor Recall that if $V$ is a vector space, then a metric tensor of $V$ is an element $g \in V^* \otimes V^*$, that is symmetric and positive definite, i.e. * (symmetric) for any $v, w \in V$, $g(v,w) = g(w, v)$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-07&rev=1771684876&do=diff 2020-02-07, Friday Today, we talk about Laplacian operator. The formula Take $\R^n$ to be our space. Let $x_1, \cdots, x_n$ be the standard coordinates. Let $u_1, \cdots, u_n$ be a general curvilinear coordinates. Then, we have vector basis transformation rule $$ \frac{\d}{\d u_i} = \sum_j \frac{\d x_j}{\d u_i}\frac{\d}{\d x_i}. $$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-10&rev=1771684876&do=diff 2020-02-10, Monday $$\gdef\div{\text{div}} \gdef\vol{\text{Vol}} \gdef\b{\mathbf} \gdef\d{\partial}$$ We first finish up the derivation of Laplacian last time. See the lecture note 2020-02-07, Friday. Then, we review three concepts $df, \nabla f, \nabla \cdot \b V$ ($\nabla \times \b V$ is a bit special for $\R^3$). text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-14&rev=1771684876&do=diff 2020-02-14, Friday We will follow Chapter 13 in Boas. After introducing some most important equations: heat equation, wave equation, Schroedinger equation, Laplace equation, we will do an exercise on separation of variable. Questions to ponder: text/html 2026-02-21T14:41:17+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-19&rev=1771684877&do=diff 2020-02-19, Separation of Variables The idea (equation only, without boundary condition) Suppose you have a differential operator $P(x,y) = P_1(x) + P_2(y)$, and you have a differential equation about $F(x,y)$ $$ P(x,y) F(x,y) = 0 $$ Then one can try to find a basis $\{ F_n(x,y)\}$ for the solution space, where each $F_n$ can be written as (omitting the subscript $n$) $$ F(x,y) = A(x) B(y)$$ Then the equation become $$ B(y) P_1(x) A(x) + A(x) P_2(y) B(y) = 0 $$ Divide both sides by $A(x) B(… text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-21&rev=1771684876&do=diff 2020-02-21, Friday Today we finish the discussion of separation of variables, then we review for the midterm 1. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:02-26&rev=1771684876&do=diff 2020-02-26, Wednesday Today we begin Chapter 12, the series solution to ODE. Consider a linear differential equation $$ P(x) y(x) = 0 $$ where $P(x)$ is some differential operator in $x$. The philosophy is that, assume the solution exists and is analytic around $x=0$, namely, we can do Taylor series of $y(x)$ around $x=0$, then $$ y = \sum_{n=0}^\infty a_n x^n $$ then, we can try to figure out what is the relations between $a_n$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:03-11&rev=1771684876&do=diff 2020-03-11, Wednesday Today we talked about section 13-15, the second Bessel function, an aside on Gamma function, the shape of Bessel functions, and some recursion relations. The explanation from Boas are quite detailed, so I won't repeat it here. A rough outline of the lecture is the following text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:03-20&rev=1771684876&do=diff 2020-03-20, Friday Today we will continue discussing about the steepest descent method, and its application to finding the asymptotic behavior of special functions. Bessel function Recall Bessel function with integer order has the following integral expression. $$ J_n(x) = \oint_{|u|=1} u^{-1-n} e^{\frac{x}{2}(u - 1/u)} \frac{du}{2\pi i} $$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:03-30&rev=1771684876&do=diff 2020-03-30, Monday Today, we finish up some loose ends in Chapter 12 and talk about a few exercises. Other Kinds of Bessel Functions (Boas 12.17) Speherical Bessel function $j_n(x), y_n(x)$ These are related to half-integer order Bessel functions $J_{n+1/2}(x), Y_{n+1/2}(x)$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:04-01&rev=1771684876&do=diff 2020-04-01, Wednesday Recall the first time we dipped in Chapter 13, partial differential equations, we discussed about the 'method of separation of variables'. We covered Boas 13.1 and 13.2. Today, we plan to talk about 13.3 and 13.4, the heat equation, Schroedinger equation and the wave equation. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:04-03&rev=1771684876&do=diff 2020-04-03, Friday Today we consider PDE problem in cylindrical coordinate and spherical coordinate. Cylindrical Coordinate Recall that Laplacian in cylindrical coordinate $r, \theta, z$ is written as $$ \Delta u = \frac{1}{r} \d_r(r \d_r(u)) + \frac{1}{r^2} \d_\theta^2 u + \d_z^2 u. $$ We shall look for eigenfunctions of $\Delta$ of the following form $$ u(r,\theta, z) = R( r ) \Theta(\theta) Z(z), $$ Then we have $$ \frac{1}{u}\Delta u = \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) + \frac{1}… text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:04-06&rev=1771684876&do=diff 2020-04-06, Monday Today, we consider those PDEs with Laplacian in spherical coordinate. Eigenfunction of Laplacian in spherical coordinate. $$ \Delta u = \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d u}{\d r} \right) + \frac{1}{r^2} \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d u}{\d \theta}) + \frac{1}{r^2 \sin^2 \theta} \frac{\d^2 u}{\d \phi^2}. $$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:04-08&rev=1771684876&do=diff 2020-04-08, Wednesday Today we consider Poissson equation, for example, solving for the gravity potential function when there exists a source distribution $$ \nabla u(x,y,z) = f(x,y,z), \quad u(x,y,z) \to 0, \z{ as } |(x,y,z)| \to \infty. $$ Green's function text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:04-15&rev=1771684876&do=diff 2020-04-15, Wednesday Concepts There is nothing you cannot illustrate by drawing two circles. Sample Space : just a set, usually denoted as $\Omega$. Elements of $\Omega$ are called outcome. Certain nice subsets of $\Omega$ are called events. (In practice, all subsets that you can cook up are 'nice'.) text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:10.7&rev=1771684876&do=diff Problem 10.7 Show that $$ \frac{d^{l-m}}{dx^{l-m}} (x^2-1)^{l} = \frac{(l-m)!}{(l+m)!} (x^2-1)^m \frac{d^{l+m}}{dx^{l+m}} (x^2-1)^l $$ Solution We first rearrange the factorials, so that the differential operators looks nicer $$ \tag{*} \frac{1}{(l-m)!} \frac{d^{l-m}}{dx^{l-m}} (x^2-1)^{l} = (x^2-1)^m \frac{1}{(l+m)!} \frac{d^{l+m}}{dx^{l+m}} (x^2-1)^l $$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:about_vectors_and_tensors&rev=1771684876&do=diff Exercises about Vectors and Tensors If you are unfamiliar with the sentence “elements of a set” or “elements of a vector space”, you can take a look at Sets. Getting Familiar with basis free notation The whole point of doing abstract vector space, is to show that, we can do linear algebra without using the crutches of 'basis'. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:ex2&rev=1771684876&do=diff See the [chapter on dual vector space] for notations. My canonical pairing is using pointed bracket $\langle -, - \rangle$, Halmos uses $[-,-]$. Solution 1. $y$ is a linear function in case (a) and (b). Case (d), $y$ is not taking value in $\R$, but in $\C$. we are considering here a vector space over $\R$, hence linear functional should be valued in $\R$ as well. Case (e), this is an example of homoegeous function, that is, if you have a positive number $c$, then $y(c x) = c y(x)$, however, … text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:ex3&rev=1771684876&do=diff Exercises on Tangent Vectors and Metric Tensor Metric Tensor. Length, Area and Volume element 1. True or False * Each vector space comes equipped with a preferred inner product. * Each vector spaces comes equipped with a preferred basis. * text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:ex4&rev=1771684876&do=diff Exercises about General Coordinates We will take exercises from Boas 10.8. Try to find the $ds^2$ (same as the metric tensor). The scale factor $H_i$ (not to be confused with my symbol for the dual basis), the volume or the area element, and the $\bf a$ factor. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:final-sol&rev=1771684876&do=diff Final Solution $$\gdef\E{\mathbb E}$$ Due Date : May 10th (Sunday) 11:59PM. Submit online to gradescope. Policy : You can use textbook and your notes. There should be no discussion or collaborations, since this is suppose to be a exam on your own understanding. If you found some question that is unclear, please let me know via email. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:final&rev=1771684876&do=diff Final $$\gdef\E{\mathbb E}$$ Due Date : May 10th (Sunday) 11:59PM. Submit online to gradescope. Policy : You can use textbook and your notes. There should be no discussion or collaborations, since this is suppose to be a exam on your own understanding. If you found some question that is unclear, please let me know via email. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:home&rev=1771684876&do=diff Math 121B: Mathematics for Physical Sciences UC Berkeley, Spring 2020 Zoom Meeting: <https://berkeley.zoom.us/j/486973340> MWF 9-10AM Zoom Office hours: MWF: 10-11am, and Monday Friday afternoon 4-5pm. * My Personal Meeting ID is: 881-910-2324. * Please also join the chat channel, named text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:midterm2-solution&rev=1771684876&do=diff Midterm 2, Solution 1. Gamma functions and Beta function (15 points) You may use Gamma or Beta function to express the final answer. Please show intermediate steps, otherwise there are no points. 1. Compute the integral (5 points) $$\int_0^1 \frac{x^4}{\sqrt{1-x^3}} dx $$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:midterm2&rev=1771684876&do=diff Midterm 2 for Math 121B Please write or type your solution, and upload to gradescope. Due date: 4/12 Sunday 11:59pm. You may use book and your note, but no discussion with each other. 1. Gamma functions and Beta function (15 points) You may use Gamma or Beta function to express the final answer. Please show intermediate steps, otherwise there are no points. text/html 2026-02-21T14:41:17+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:prob9-1&rev=1771684877&do=diff Chapter 12 Problem 9.1 Expand the following function into Legendre series. $$ f(x) = \begin{cases} -1 & -1 < x < 0 \cr 1 & 0 < x < 1 \end{cases} $$ Solution: We need to compute $$ c_n = \frac{2n+1}{2} \int_{-1}^1 f(x) P_n (x) dx $$ Then we can find that $$ f(x) \approx \sum_{n=0}^\infty c_n P_n(x). $$ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:r&rev=1771684876&do=diff Programming in R Cheat Sheat and Examples Cheat Sheat reference: [ CheatSheat PDF] variable manipulations function meaning c(1,3,4,5) create a vector 2:5 create a sequence (2,3,4,5) y[3] say y is a vector, take the 3rd entry out of y seq(2,10, by=0.5) create a sequence, start from 2, ending at 10, step size = 0.5 text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:sample-m1&rev=1771684876&do=diff sample Midterm 1 1. Let $V = \R^2$, and let $v = (3,2)$ in the Cartesian basis of $\R^2$. Now, we choose another basis as follows $$ e_1 = (2,1), \quad e_2 = (0,1) $$. Expand $v$ in terms of $e_1, e_2$. 2. Let $V = \{a + bt + ct^2 \mid a, b, c \in \R\} $ be the space of polynomials of degree at most 2. Let $f_1(t), f_2(t), f_3(t)$ be three elements in $V$, given as follows $$ f_1(2) = 1, \quad f_1(3) = 0, \quad f_1(5) = 0 $$ $$ f_2(2) = 0, \quad f_2(3) = 1, \quad f_2(5) = 0 $$ $$ f_3(2) = 0, … text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:sets&rev=1771684876&do=diff Sets A vector space is a set with certain properties. What is a set? A set is a collection of elements. For example, we can say, “the set of students in this room”, where each student would be an element of this set. Or we can say, “the set of possible outcomes of tossing a coin text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:start&rev=1771684876&do=diff 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 text/html 2026-02-21T14:41:17+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math121b:tmp&rev=1771684877&do=diff Tensor Analysis for physics and engineering students There are two ways to introduce Laplacian in curvilinear coordinate, one is following Boas, one is following the standard math language. I will present both approaches. Curvi-linear coordinate