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UW Courses Instructions
水卢课程指南
Instructor: Sibelius
Last update: January 3, 2019

Copyright ©2017–2019 Sibelius Peng
Permission is NOT granted to copy, distribute and/or modify this document.

Contents

I.

MATH

5

1. Basic Math
1.1. Calculus . . . . . . . . . . . . . .
1.1.1. other faculties’ calculus .
1.1.2. MATH 137 . . . . . . . .
1.1.2.1. Topics . . . . .
1.1.2.2. Final Note . . .
1.1.3. MATH 138 . . . . . . . .
1.1.3.1. Topics . . . . .
1.1.3.2. Final Note . . .
1.1.4. MATH 147 . . . . . . . .
1.1.4.1. Topics . . . . .
1.1.5. MATH 237 . . . . . . . .
1.1.5.1. Selected Proofs
1.2. Algebra . . . . . . . . . . . . . . .
1.2.1. other faculties’ algebra .
1.2.2. MATH 135 . . . . . . . .
1.2.2.1. Topics . . . . .
1.2.3. MATH 136 . . . . . . . .
1.2.3.1. Topics . . . . .
1.2.4. MATH 145 . . . . . . . .
1.2.4.1. Topics . . . . .

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2. STAT
2.1. STAT 220/230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. ACTSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
15

3. AMATH
3.1. AMATH 231 . . . . .
3.1.1. Topics . . . . .
3.1.2. Selected Proof
3.2. AMATH 251 . . . . .
3.2.1. Topics . . . . .
3.3. AMATH 390 . . . . .
3.3.1. Topics . . . . .
3.4. AMATH 391 . . . . .

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Contents
4. PMATH

31

5. CO
5.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32
33

II. CS

34

6. Basic
6.1. CS 135 . . . . . .
6.1.1. Overview
6.2. CS 136 . . . . . .
6.2.1. Overview

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35
35
35
35
35

7. Major
7.1. CS 246 . . . . . .
7.1.1. Summary
7.2. CS 251 . . . . . .
7.2.1. Topics . .
7.2.2. Diary . .

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36
37
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37

8. Minor

38

9. Beyond

39

III. MUSIC

40

10.Music Theory
10.1. MUSIC 111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41

11.Music Ensemble

42

IV. OTHER

44

12.Science

45

13.Arts

46

4

Part I.

MATH
1. Basic Math

7

1.1. Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.1.1. other faculties’ calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.1.2. MATH 137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.1.2.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.1.2.2. Final Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.3. MATH 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.3.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.3.2. Final Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.1.4. MATH 147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.1.4.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.1.5. MATH 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.1.5.1. Selected Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2. Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.2.1. other faculties’ algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.2.2. MATH 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2.2.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2.3. MATH 136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2.3.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2.4. MATH 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2.4.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2. STAT

15

2.1. STAT 220/230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2. ACTSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3. AMATH

16

3.1. AMATH 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.1.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.1.2. Selected Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.2. AMATH 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

TABLE OF CONTENTS
3.3. AMATH 390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.3.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.4. AMATH 391 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

4. PMATH

31

5. CO

32

5.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

33

1
Basic Math

顾名思义，这部分主要是一些数学的基础课。水卢的基础课划分的比国内清楚一些。国
内大学有些时候把国内的基础课直接合并在一起，把数学分析和高代作为基础，其中其实包
括很多分支。我们这边一般把基础课分作微积分和代数两个 (Calculus and Algebra) 。当然
往后面学完全看个人兴趣和专业需求，接下来我们就来详尽分析一下这些基础课吧。
为了连续性，我这里不会按照学期的顺序，而是按照课程的连贯性，比如我会将微积分
单独作为一个部分来详尽阐述。

1.1. Calculus
什么是微积分呢？这个问题大概在学 math 237 的时候 Professor Spiro Karigiannis 讲
了一下 — How things change. 就是事物如何变化的。学了一年微积分也不明白学的是什
么. . . . . .

1.1.1. other faculties’ calculus
详尽的课程介绍可以自己去官网或者uwflow进行查找。这个课我也没上过，不过听了其
他系的人上的人说也不难. . . . . .
MATH 104 Introductory Calculus for Arts and Social Science
MATH 116 Calculus 1 for Engineering
MATH 117 Calculus 1 for Engineering
上面两门课虽然名字一样，但是受众不一样。116 Open to students in Engineering
excluding Electrical and Computer Eng, Nanotechnology Eng, Software Eng and Systems
Design Eng. 117 Open only to students in Electrical and Computer Engineering or Software
Engineering or Nanotechnology Engineering.
MATH 118 Calculus 2 for Engineering
MATH 119 Calculus 2 for Engineering
跟上面一样的，也是受众不一样。118 Open only to students in Engineering excluding
students in Electrical and Computer Eng, Nanotechnology Eng, Software Eng and Systems
Design Eng. 119 Open only to students in Electrical and Computer Engineering or Software
Engineering or Nanotechnology Engineering.

7

1. Basic Math
MATH 124 Calculus and Vector Algebra for Kinesiology.
这个专业其实是很有意思的一个专业，有兴趣的同学可以研究一下。
MATH 127 Calculus 1 for the Sciences.
虽然是 for Science, 很多数学系的学不下去137也会来上127，而且也是可以接受的（具
体的请查询calendar 或者询问advisor）。
MATH 128 Calculus 2 for the Sciences.
跟上面一样，很 137 failed 的人或者说达不到138的要求就会来上这个课，其实难度上并
无太大差别。

虽然其他有些低级别的课也可以是 math major 的课，不过正常来说接下来才是真正的
数学系的课。

1.1.2. MATH 137
这门课对于作为大学第一门入门微积分课一定要好好学，为以后的知识打好基础。其实
我觉得这门课太简单了，没什么好多说的，不过还是多说一下吧，因为很多新生刚来难以适
应大学的课程模式和结构，当然大佬除外。
这门课之前是由作业构成的，然后也一直在2017 fall 改版了，增加了proof的部分，当然
难度跟math147当然不能比。2018 fall 把原有的作业全部改成weekly quiz。虽然难度不大，
但是quiz的目的就是督促人及时复习，而不是拖到考试再复习。
1.1.2.1. Topics
• Absolute Values
• Sequence and their limits
• Squeeze Theorem, Induction, MCT
• Function and Limits
• Infinite Limits, Continuity
• IVT, EVT, Instantaneous Velocity
• Derivatives, Differentiation Rules
• Tangent Lines, Newton’s Method, IFT
• Implicit Differentiation, Extrema, MVT
• Application of MVT, l’Hopital’s Rule
• Curve Sketching
• Taylor Polynomials and Taylor’s Theorem
最难的应该就是泰勒那里了，但是实际上不需要很懂原理，会用就好了。很大可能
最后有些prof讲得慢，然后泰勒后面的一些定理就不考了，还有big O notation等等，但
是Jordan讲得还是很快的，所以应该能在他的课上学习到这些不会考的知识点。
还有一点就是证明要写清楚，不要伪证或者跳步骤。本来就不是很难的证明，一定要尽
量拿满分。

8

1. Basic Math
1.1.2.2. Final Note
这门课的好prof应该挺多的，不过elas分配的prof据说是真的差，上课都不能自圆其
说，然后Jordan讲得又太好了，导致很多人都去蹭Jordan的课。如果教室大还好，教室
小了很多enroll的人就只能站着，所以很多人在第一学期蹭课就遭遇查ID的情况。我建议
最好能抢到他的课，抢不到的话就蹭吧，查ID的话就蹭别的好prof吧，像Eddie等等。当
然Jordan的review session还是非常值得去的，很多情况下都是类似的题目，如果有机会蹭的
话可以去蹭一下。
这门课还有一个physics-based section，很多地方都应用到物理上了。如果不感兴趣物理
不建议上这个。当然有些人可能专业需要，那还是上吧，难度也不会大的离谱。

1.1.3. MATH 138
传说中难度陡然增大的课。相比137确实难了不少。1 137很多同学都在吃老本，138讲积
分很多同学也吃一点老本，后面讲级数series就没得吃了。下个学期怎么改我不太清楚，我
就先说说我这个学期吧。
1.1.3.1. Topics
• Area Under Curves, Riemann Sums, Definite Integrals
• Definite Integrals, Average Value, FTC I, FTC II
• Change of Variable, Trigonometric Substitution, Integration By Parts (IBP)
• Partial Fractions, Improper Integrals
• Areas, Volumes, Arclength
• DEs, IVPs
• Qualitative Solutions
• Series
– Convergence
– Integral Test
– Ratio Test
– Alternating Series
– Positive Series
– Binomial Series
– Taylor Series
– Power Series
• Curves
• Parametrization
所以这门课两大theme： Integrals & Series. 积分有的很需要技巧，然后DE的难度是很
小的，因为只是intro，最后series的难度会陡然增大，建议就是多做题吧，然后考试应该就
还好了。

1

当然有人也觉得是很简单的。

9

1. Basic Math
1.1.3.2. Final Note
很多人上完这门课，微积分就结束了，因为general cs不需要后面的微积分了。微积分
相比线代对cs的作用不是那么大，但是还是看选择的cs方向。如果选择computer graphics的
话，建议接着往下学微积分（math237）。cs370（Numerical Computation）是cs488(computer
graphics)的preq。如果要上cs370，微积分的底子就显得尤为重要了。
很久以前cs是必修4门微积分的（amath231）2 。 后来cs考虑到要上更多的cs课，就把微
积分必修的要求减轻了，只用两门了。当然从现在的水平来看，微积分确实没有很大的用
处，所以先学着备着吧，假如将来真的做computer graphics，微积分的用处就显现出来了。
再说一个用处，当然这是建立在对数学很感兴趣的前提下，如今很多人对数学也就
是专业需要就学一下，并没有很大兴趣。好了，进入正题。pmath365（differential geometry，微分几何）也是很有趣的一个数学subject，preq：amath231/math247，这是很需要微
积分做底子的，不过更多的是从分析、理论的层面上，不是像math137/138 或者amath等等
从应用的层面上。学完这个，就能感受到数学之美了3 。还有pmath467（Algebraic Topology），preq：pmath347/351。这个偏代数，但是学起来一定是很有趣的。
还有以后如果想学习real/complex analysis, 微积分也是尤为重要的。当然如果137/138上
过来，要pmath333过渡一下到pmath351/352，或者图简单，上pmath331/332也行，但是这
个简单的版本偏应用，理论相对较少，所以难度也是降低了很多。
如果从amath专业来考虑的话，微积分要学好，然后real/complex analysis学baby version 就够了，多学点应用便好。相反的，pmath专业或者enthusiasts 学351/352 就比较好
了，见识和学习到更多数学好玩的地方。

1.1.4. MATH 147
1.1.4.1. Topics
• The real numbers
• Sequences and limits
• Logarithms, exponentials and other important functions
• Functions, limits and continuity
• Intermediate Value and Extreme Value Theorems
• Derivatives and curve sketching
• The Mean Value Theorem and applications
• Taylor’s theorem

1.1.5. MATH 237
1.1.5.1. Selected Proofs
Theorem 1.1.1
⃗, then f is differentiable at a
⃗.
Suppose f is in C 1 at a
Proof
▸ (for n = 2, but the proof is identical for any n.)
⃗ = (a, b).
Denote the function by f (x, y), and the point a
2
3

一个amath prof告诉我的. . . . . .
我将来打算上，不知道有没有机会[捂脸]

10

1. Basic Math
Our hypothesis: fx , fy both exist near (a, b) and are continuous at (a, b)
The linear approximation is
⃗) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)
L(x, y) = f (⃗
a) + (∇f )(⃗
a) ⋅ (⃗
x−a
We want to show

∣f (x, y) − L(x, y)∣
√
=0
(x,y)→(a,b)
(x − a)2 + (y − b)2
lim

f (x, y) − L(x, y) = f (x, y) − f (a, b) − fx (a, b)(x − a) − fy (a, b)(y − b)
= f (x, y) − f (a, y) + f (a, y) − f (a, b) −fx (a, b)(x − a) − fy (a, b)(y − b)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
fx (x0 ,y)(x−a) by (2)

fy (a,y0 )(y−b) by (3)

(1)
Consider the function
h(t) = f (t, y)

on [a, x]

(y is fixed )

h′ (t) = fx (t, y). By hypothesis, fx exists near (a, b), so h is differentiable on [a.x] fo x close
to a.
Apply MVT to this situation: there exists x0 between a and x such that
h( x ) − h( a ) = h′ ( x0 )(x − a)
®
®
¯ ±
t2

t1

t0

t2 −t1

f (x, y) − f (a, y) = fx (x0 , y)(x − a)

(2)

Consider the function
k(s) = f (a, s)

on [y, b]

(a is fixed )

k ′ (s) = fy (a, s) exists near (a, b), so k is differentiable on [y, b] for y close to b.
Let s1 = y, s2 = b. Apply MVT, there exist y0 between b and y such that
k(s0 ) − k(s1 ) = k ′ (y0 )(s2 − s1 )
f (a, y) − f (a, b) = fy (a, y0 )(y − b)

(3)

We’ve shown that: there exists x0 between a and x and there exists y0 between b and y such
that
(x − a)[fx (x0 , y) − fx (a, b)] (y − b)[fy (a, y0 ) − fy (a, b)]
f (x, y) − L(x, y)
√
√
√
=
+
2
2
(x − a) + (y − b)
(x − a)2 + (y − b)2
(x − a)2 + (y − b)2
We want to show this → 0 as (x, y) → (a, b).
∣A + B∣ ≤ ∣A∣ + ∣B∣ also
∣x − a∣
√
≤1
(x − a)2 + (y − b)2

√

11

∣y − b∣
(x − a)2 + (y − b)2

≤1

1. Basic Math
Then

f (x, y) − L(x, y)
√
≤ ∣fx (x0 , y) − fx (a, b)∣ + ∣fy (a, y0 ) − fy (a, b)∣
(x − a)2 + (y − b)2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
→ 0 as (x,y)→(a,b)
since fx is continuous
at (a,b)

→ 0 as (x,y)→(a,b)
since fy is continuous
at (a,b)

So by Squeeze Theorem
RRR
R
RRR √f (x, y) − L(x, y) RRRRR = 0
R
R
(x,y)→(a,b) RRR (x − a)2 + (y − b)2 RRR
R
R
lim

So f is differentiable at (a, b)
ˆ
Example Prove that
Proof
▸

∞

2

e−x dx converges, and find the value N .
0

ˆ

∞

2N =

e

−x2

−∞
∞

ˆ

ˆ
dx =
ˆ

2

4N 2 =

∞

2

e−y dy
−∞
∞
−y 2

e−x dx
e dy
−∞
−∞
ˆ ∞ ˆ ∞
2
2
e−x dx] e−y dy
[
=
−∞
ˆ−∞
∞ˆ ∞
2
2
=
e−(x +y ) dA
¨−∞ −∞
2
2
=
e−(x +y ) dA
R2

0≤r≤∞
0 ≤ θ ≤ 2π

Change to polar coordinates {
ˆ

2π

ˆ

∞

4N 2 =
0

ˆ

= 2π
0

0
∞

2

e−r r drdθ
´¹¹ ¹ ¸¹ ¹ ¹ ¶
dA

2
1 2 ∞
e−r r dr = 2π (− e−r )∣ = π
2
0

√
π
Hence N =
2

1.2. Algebra
1.2.1. other faculties’ algebra
详尽的课程介绍可以自己去官网或者uwflow进行查找。这个课我也没上过，不过听了其
他系的人上的人说也不难. . . . . .
MATH 103 Introductory Algebra for Arts and Social Science
MATH 106 Applied Linear Algebra 1
MATH 114 Linear Algebra for Science

12

1. Basic Math
MATH 115 Linear Algebra for Engineering

1.2.2. MATH 135
1.2.2.1. Topics
To develop the vocabulary, techniques and analytical skills associated with reading and
writing proofs, and to gain practice in formulating conjectures and discovering proofs. Emphasis will be placed on understanding basic logical structures, recognition and command over
common proof techniques, and precision in language. These skills will be developed through
working with number theory, complex numbers and polynomials.

1.2.3. MATH 136
1.2.3.1. Topics
• Vectors in Rn , Spanning
• Linear Independence, Bases, Subspaces
• Dot/ Cross Product, Projections
• Matrix
• Linear Mappings
• Vector Spaces
• Dimension
• Matrix Inverse
• Determinants
• Cramer’s Rule
• Similar Matrices
• Eigenvalues, Diagonalization
• Power of Matrices

1.2.4. MATH 145
1.2.4.1. Topics
• Chap 1. Sets and Mathematical Statements
• Chap 2. Mathematical Proof
• Chap 3. Rings, Fields and Orders
• Chap 4. Recursion and Induction
• Chap 5. Factorization of Integers
• Chap 6. Congruence and Modular Arithmetic
• Chap 7. Cryptography

13

1. Basic Math
• Chap 8. Complex Numbers
• Chap 9. Cardinality
• Chap 10. Factorization in Rings.

14

2
STAT

这 部 分 是 很 多 人 不 喜 欢 的......因 为stat不 像 其 他 数 学 那 么 有 趣 （ 可 能 以 后 会 改 变 看
法）。整个stat的课给人的感觉就是很多概念讲不明白，讲不透，但是会做题，套公式
就可以了，明白什么题套什么模型，这门课也就完了。当然如果真的对这个领域感兴趣，可
以深入研究，有很多好的课本，像概率论之类的可以自行研究，来弥补上课讲不透的东西。

2.1. STAT 220/230
这 两 门 课 我 干 脆 丢 在 一 起 了 ， 好 像 并 无 太 大 差 别 ， 可 能220相 对 简 单 一 些 。230作
为stat的第一门入门课，很多人把它当作水课。但是从这几个学期开始（18winter），这
门课的难度逐渐加大了，考试的内容也在变难，有些出的题目非常恶心，不是那种很容易就
做出来的，跟几年前学长学姐说的水课完全不一样。

2.2. ACTSC
这部分留给学姐来写好啦，因为这部分我感兴趣的也不多，也没有机会上了。

15

3
AMATH

3.1. AMATH 231
3.1.1. Topics
Before you brush your teeth,
parametrize your curves.
Edward R. Vrscay

• Vector Calculus
– gradient vector field
– Conservation in physics
– line (path) integral

ˆ

ˆ

b

f ds =

f (g(t))∥g′ (t)∥ dt

C

a

ˆ

W=

ˆ

F ⋅ dx =
C

b

F(g(t)) ⋅ g′ (t)dt
a

– Path-independence and the Fundamental Theorems of Calculus for Line Integrals
ˆ
ˆ
⃗ ⋅ dx = f (B) − f (A)
F ⋅ dx =
∇f
CAB

CAB

– First Fundamental Theorem for Line Integrals
ˆ x
⃗ (x) = F(x)
f (x) =
F ⋅ dy Ô⇒ ∇f
x0

– over closed curves

‰
F ⋅ dx
‰

ˆ
F ⋅ dx =

C

C

‰

b

F(g(t)) ⋅ g (t)dt =
′

f ds

a

̂
where the scalar valued function f (g(t)) = F(g(t)) ⋅ T(t)

16

3. AMATH
– Green’s Theorem
ˆ

¨
F ⋅ dx =

(

∂D

D

∂F2 ∂F1
−
) dA =
∂x
∂y

– divergence
⃗ ⋅F=
divF = ∇

¨
(∇ × F)z dA
D

∂F1 ∂F2 ∂F3
+
+
∂x
∂y
∂z

– divergence of position vector
⃗⋅
∇
– curl

1
r=0
r3

(x, y, z) ≠ 0

⃗ ×F=(
curlF = ∇ × F = ∇

∂F2 ∂F1
−
)k
∂x
∂y

∇ × F = 0 Ô⇒ F is gradient
– vorticity
v(x, y) = (v1 (x, y), v2 (x, y)), then vorticity is
Ω(x, y) =

∂v2 ∂v1
−
∂x
∂y

– Mean value theorem for (double) integrals
f (x , y ) = f D
∗

∗

1
=
A(D)

¨
f (x, y)dA
D

1
[curlF(p1 , p2 )]z = Ω(p1 , p2 ) = lim 2
ε→0 πε

‰
F ⋅ dx
Cε

– Total outward flux
ˆ b
ˆ b
‰
′
̂
̂
̂ =
F(g(t)) ⋅ N(t)∥g
(t)∥dt
F(g(t)) ⋅ N(t)ds
=
F ⋅ Nds
C

a

a

– Divergence Theorem
‰
¨
¨
∂F1 ∂F2
̂
⃗ ⋅ F dA
[
F ⋅ Nds =
+
] dA =
∇
∂y
D
C
D ∂x
‰
1
̂
⃗
∇ ⋅ F(p) = lim 2
F ⋅ Nds
ε→0 πε
Cε
¨
1
̂
⃗
∇ ⋅ F(p) = lim 4 3
F ⋅ Nds
ε→0 πε
S
ε
3
– Circulation integrals/Outward flux integrals around singularities
– Surface integration
∗ Surface parametrizations
∗ Normal vectors to surfaces from parameterizations
N(u0 , v0 ) = ±Tu (u0 , v0 ) × Tv (u0 , v0 )
where Tu =

∂g(u,v)
∂u .

17

3. AMATH
∗ surface integrals of scalar functions
ˆ
¨
f dS =
f (g(u, v))∥N(u, v)∥dudv
S

u,v

∗ surface area

¨
S=

∥
Duv

∗ flux integral

¨

̂
F ⋅ NdS
=

∂g ∂g
×
∥dudv
∂u ∂v

¨

S

F(g(u, v)) ⋅ N(u, v)dA
Duv

∗ Gauss Divergence Theorem in R3
˚
¨
̂
divF dV
F ⋅ NdS =
S

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

surface integral

D

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
volume integral

– Let U (t) denote the total amount of substance X in region D at time t. u(x, t)
denote the concentration of substance X at a point x ∈ R ⊂ R3 .
˚
˚
∂u
′
U (t) =
u(x, t)dV Ô⇒ U (t) =
(x, t)dV
D
D ∂t
– The flow of X is defined by the flux density vector j(x, t). The net outward substance X through an infinitesimal element of surface area ¨
dS centered at point
̂
̂
x ∈ ∂D is given by j(x, t) ⋅ N(x)dS.
The total outward flux:
j(x, t) ⋅ N(x)dS.
By Divergence Theorem, we have
˚
¨
̂
⃗ ⋅ j(x, t)dV
∇
j(x, t) ⋅ N(x)dS =
˚
and U ′ (t) = −
for substance X:

D

D

∂D

⃗ ⋅ j(x, t)dV , then integral form of the conservation law
∇
˚
(
D

– ∇ operator

∂D

∂u
⃗ ⋅ j(x, t)) dV = 0
(x, t) + ∇
∂t

RRR i
j
k RRRR
RRR
RRR
RRR
RR
RRR ∂
∂
∂ RRR
∇ × F = RR ∂x ∂y
∂z RRR
RRR
RRR
RRR
RRRF F F RRRRR
2
3R
R 1
∂u ∂u ∂x ∂u ∂y
∂x
=
+
= ∇u ⋅
∂v1 ∂x ∂v1 ∂y ∂v1
∂v1

∇ in polar/cylindrical/spherical coordinates in R3
– Stoke’s Theorem

ˆ

¨
F ⋅ dx =

C

̂
(∇ × F) ⋅ NdS

S

– final calculation (using spherical polar coordinate is easiest)
F=

1
3−n
1
⃗ ⋅F=
[xi + yj + zk] Ô⇒ ∇
r= 2
n
2
2
n/2
r
rn
(x + y + z )

18

3. AMATH
• Fourier Series
f (x) =

∞
a0 ∞
+ ∑ an cos nx + ∑ bn sin nx,
2 n=1
n=1

x ∈ [−π, π]

– piecewise C 1
– pointwise convergence for a Fourier series (2π-period and piecewise C 1 )
– even & odd extension of a function defined on (0, π)
–
⟨fn , fn ⟩ =

n
a20
π + π ∑ [a2k + b2k ] ≤ ∥f ∥22
2
k=1

– {fn } converges pointwise to f : lim ∣fn (x) − f (x)∣ = 0, for all x ∈ [a, b].
n→∞

∞

– Series ∑ ak in C[a, b] converges uniformly to f means that the sequence {Sn } of
k=1

partial sums converges uniformly to f
∞

lim ∥f − ∑ ak ∥

n→∞

k=1

=0
∞

∞

– Series ∑ ak in C[a, b] converges in the mean to f means that the sequence {Sn }
k=1

of partial sums converges in the mean to f
∞

lim ∥f − ∑ ak ∥ = 0

n→∞

k=1

2

– Weierstrass M-test (also covered in MATH 148): If ∣an (x)∣ < Mn ,
∞

∞

n=1

n=1

∀x ∈ [a, b],

∑ Mn converges, then ∑ an (x) converges absolutely for each x ∈ [a, b], with sum
f (x), it converges uniformly to f (x) on [a, b].

–

∗ fp piecewise continuous Ô⇒ converges in the mean to fp on any finite interval
∗ fp piecewise C 1 Ô⇒ converges pointwise to fp for all x ∈ R
∗ fp piecewise C 1 and continuous Ô⇒ converges uniformly to fp on any finite
interval

– Complex Fourier series of a τ-periodic function
ˆ τ
∞
2
2
f (t)2 dt = τ ∑ ∣cn ∣2
– Parseval’s formula: ⟨f, f ⟩ = ∥f ∥2 =
− τ2

– Fourier Transform:

−∞

ˆ

∞

F{f (t)} = F (ω) =

f (t)e−iωt dt
−∞

3.1.2. Selected Proof
Conservation of Energy Assumptions: F ∶ Rn → Rn is conservative.
Proof
▸
1
E(t) = m∥v(t)∥2 + V (x(t))
2

19

3. AMATH
E ′ (t) =

md
d
[v(t) ⋅ v(t)] + V (x1 (t), . . . , xn (t))
2 dt
dt

d
[v(t) ⋅ v(t)] = v′ (t) ⋅ v(t) + v(t) ⋅ v′ (t) = 2a(t) ⋅ v(t)
dt
By chain rule:
d
∂V dx1
∂V dxn
⃗ ⋅v
V (x1 (t), . . . , xn (t)) =
+ ... +
= ∇V
dt
∂x1 dt
∂xn dt
Put together,
⃗ = v(t) ⋅ [ma(t) + ∇V
⃗ ] = v(t) ⋅ [ma(t) − F(x(t))] = v(t) ⋅ 0 = 0
E ′ (t) = v(t) ⋅ [ma(t)] + v(t) ⋅ ∇V

Work Integrals
Proof
▸

⃗
F = −∇V

ˆ
W=

ˆ

⃗ ⋅ dx = −[V (B) − V (A)] = V (A) − V (B) = −∆V
∇V

F ⋅ dx =
CAB

CAB

Second Fundamental Theorem of Line Integrals Let F ∶ U → Rn be a continuous vector
field on a connected open set U ⊂ Rn , and let x1 , x2 be two points in U. If F = ∇f , where
f ∶ U → R is a C 1 scalar field, and C is any curve in U joining x1 to x2 , then
ˆ
F ⋅ dx = f (x2 ) − f (x1 )
C

Proof
▸ Let C be given by x = g(t), t1 ≤ t ≤ t2 , so that x1 = g(t1 ), x2 = g(t2 ). Bt the hypothesis,
ˆ
ˆ
F ⋅ dx = (∇f ) ⋅ dx
C
C
ˆ t2
=
∇f (g(t)) ⋅ g′ (t)dt
by definition of line integral
t1
t2

ˆ
=

t1

d
[f (g(t))]dt
dt

Chain rule

= f (g(t2 )) − f (g(t1 ))

second FTC

= f (x2 ) − f (x1 )

Proposition 5.1
ˆ
∥f ∥22 =

b

f (x)2 dx
a

≤ max [f (x)]2 (b − a)
a≤x≤b

2

= [max ∣f (x)∣] (b − a)
a≤x≤b

= (b − a)∥f ∥2∞

20

3. AMATH
Proposition 5.2 If fn converges uniformly in piecewise continuous [a, b], then converges in
(i) mean and (ii) pointwise. Proof
▸ For (i), use prop 5.1 and squeeze theorem. For (ii) by definition, ∣fn (x) − f (x)∣ ≤ ∥fn − f ∥∞ ,
then...

3.2. AMATH 251
3.2.1. Topics
• First-order DEs
– An equation relating an unknown function and one or more of its derivatives is
called a differential equation.
– The order of a differential equation is the order of the highest derivative that
appears in it.
– IVP & IC
– Ordinary differential equations: the unknown function (dependent variable) depends on only a single independent variable.
– If the dependent variable is a function of two or more independent variables, then
partial derivatives are likely to be involved; if they are, the equation is called a
partial differential equation.
– general & particular solution
– Slope fields & solution curves
– Theorem Existence and Uniqueness of Solutions
dy
∂f
= f (x, y), y(a) = b. has only one solution is defined on I, if f (x, y) &
are
dx
∂y
continuous on some rectangle R in the xy-plane that contains the point (a, b) in
its interior.
– Separable Equations
– Linear First-Order

dy
+ P (x)y = Q(x),
dx
´

Integrating factor: ρ(x) = e

y(x0 ) = y0

P (x)dx

– Theorem Unique solution for linear first-order equation
P (x) and Q(x) are continuous on the open interval I containing x0
– Substitution Methods
dy
y
y
∗
= F ( ) Ð→ v =
dx
x
x
∗ Bernoulli Equation
dy
+ P (x)y = Q(x)y n
dx

Ð→ v(x) = y 1−n

– Exactness (will not be tested on the final)
M (x, y) + N (x.y)

dy
=0
dx

∂M ∂N
=
⇐⇒ exact in an open rectangle R
∂y
∂x

– Reducible Second-Order Equations

21

3. AMATH
∗ Dependent variable y missing
∗ Independent variable x missing
•

Models in Chapter 1
dx
= kx
dt
= k[A(t) − T ]

– Natural Growth and Decay
– Newton’s Cooling

dT
dt

– Torricelli’s Law
Suppose that a water tank has a hole with area a at its bottom, from which water
is leaking. Denote by y(t) the depth of water in the tank at time t, and by V (t) the
volume of water in the tank then. It is plausible – and true, under ideal conditions
√
– that the velocity of water exiting through the hole is v = 2gy, which is the
velocity a drop of water would acquire in falling freely from the surface of the
water to the hole. As a consequence,
√
dV
dV
√
= −av = −a 2gy Ð→
= −k y
dt
dt

√
where k = a 2g

Alternatively, Let A(y) denote the horizontal cross-sectional area of the tank at
height y.
√
dV dV dy
dy
dy
√
=
⋅
= A(y)
Ô⇒ A(y)
= −a 2gy = −k y
dt
dy dt
dt
dt
– Mixture Problem:
dp
= Rate of change of p in time = rate pollution in − rate pollution out
dt
= (rate water in)(concentration pollutions in) − (rate water out) (concentration
pollution out)
• Mathematical Models and Numerical Methods
– Population Models:
β(t)/δ(t) - # of births/deaths per unit of per population per unit time at time t
dP
= [β(t) − δ(t)] P
dt
∗ Logistic equation
Ô⇒ P (t) =

dP
= kP (M − P ),
dt
M P0
P0 + (M − P0 )e−kM t

P (0) = P0
lim P (t) =

t→+∞

M P0
=M
P0 + 0

M : limiting population / carrying capacity
∗ A constant solution of a differential equation is sometimes called an equilibrium solution.
∗ the critical point c is stable if, for each ε > 0, there exists δ > 0 such that
∣x0 − c∣ < δ Ô⇒ ∣x(t) − c∣ < ε
for all t > 0. Otherwise it is unstable.
dx
∗ Logistic Population with Harvest
= kx(M − x) − h
dt

22

3. AMATH
– Acceleration-Velocity Models
∗ Resistance Proportional to Velocity
dv
mg
FR = −kv. m
= −kv − mg.
∣vτ ∣ =
.
dt
k
∗ Resistance Proportional to Square of Velocity
√
mg
dv
FR = −kv∣v∣.
m
= −mg − kv∣v∣.
∣vτ ∣ = v =
dt
k
GM m
– Newton’s Law of Gravitation: F =
r2
⎫
⎪
⎪
⎪
√
⎪
⎪
⎪
1 1
dv
GM
⎪
⎬ Ô⇒ v
= − 2 Ô⇒ v = v02 + 2GM ( − )
⎪
dr
r
r R
⎪
d2 r dv dv dr
dv ⎪
⎪
⎪
⎪
=
=
=v
⎪
dt2 dt dr dt
dr ⎭

GM
GM
dv d2 r
= 2 =−
=− 2
2
dt dt
(R + y)
r

Consider the interval of existence,
we must have the radicand > 0. Thus we can
√
2GM
.
find the escape velocity v =
R
– Numerical Approximation
dy
∗
= f (x, y), y(x0 ) = y0 . Step size h.
yn+1 = yn + hf (xn , yn ).
dx
∗ Improved Euler Method
k1 = f (xn , yn )
un+1 = yn + h ⋅ k1

predictor

k2 = f (xn+1 , un+1 )
1
yn+1 = y + h ⋅ (k1 + k2 )
2

corrector

• Dimensional Analysis
– Two principles
1. One can only add, subtract or equate physical quantities with the same physical
dimensions.
2. Quantities with different dimensions may be combined by multiplication with
dimensions.
– Dimensionless Variables
– Buckingham-π Theorem
Qn = f (Q1 , . . . , Qn−1 )

is equivalent to

πk = h(π1 , . . . , πk−1 )

r independent fundamental physical dimensions. k = n − r.
– Pendulum Model
• Linear Equations of Higher order
– boundary value problem / initial value problem
– Theorem Principle of Superposition for Homogeneous Equations: y = c1 y1 + c2 y2
is also a solution on I.
– Theorem Existence and Uniqueness for Linear Equations: y ′′ + p(x)y ′ + q(x)y =
f (x) has unique solution on I that satisfies y(a) = b0 , y ′ (a) = b1 .

23

3. AMATH
– homogeneous & nonhomogeneous (associated homogeneous)
– linear independence of functions
– Wronskian. Suppose the functions f1 , . . . , fn are
interval I:
⎡ f1
⎢
⎢
W (f1 , . . . , fn ) = det ⎢ ⋮
⎢ (n−1)
⎢f
⎣ 1

n − 1 times differentiable on some
...
fn ⎤⎥
⋱
⋮ ⎥⎥
(n−1) ⎥
⎥
. . . fn
⎦

f1 , . . . , fn linearly independent Ô⇒ W (f1 , . . . , fn ) ≡ 0 on I.
– Theorem General Solution for a Linear Homogeneous Equation.
y (n) + P1 (x)y (n−1) + . . . + Pn (x)y = 0

(3.1)

Let φ(x) be any solution of (3.1), y1 , . . . , yn be linearly independent solutions on
I, then there exists c1 , . . . , cn such that
n

φ(x) = ∑ ci yi (x),

∀x ∈ I

i=1

Note the difference from Superposition Theorem... I got no marks on proving this
in midterm...
Proof (n = 2) Let φ(x) be a solution of (3.1) on I. Let a ∈ I. Consider the linear
system.
φ(a)
y (a) y2 (a) c1
]( ) = ( ′ )
(∗)
[ 1′
′
y1 (a) y2 (a) c2
φ (a)
Since y1 , y2 are linearly independent on I , W (y1 , y2 ) ≠ 0 on I. Thus det(M ) ≠ 0
and (∗) has a solution
c1
φ(a)
( ) = M −1 ( ′ )
c2
φ (a)
Using these values of c1 , c2 define
y(x) = c1 y1 (x) + c2 y2 (x)
Then y(x) satisfies the IVP on I consisting of (3.1) and y(a) = φ(a), y ′ (a) = φ′ (a).
But φ(a) also satisfies this IVP on I. So by E/U we must have:
φ(x) = y(x) = c1 y1 (x) + c2 y2 (x) x ∈ I
In other words, given y1 , . . . , yn linearly independent solutions of (3.1), and arbitrary constants c1 , . . . , cn
c1 y1 (x) + . . . + cn yn (x)
is a general solution of (3.1).
– General Solution for a Linear Non-Homogeneous Equation.
– Homogeneous, linear ODEs with constant coefficients
∗ characteristic equation/polynomial: an rn + . . . + a1 r + a0 = 0
∗ Three cases: (ci are arbitrary constants)
- Linear independence verification uses Wronskian.
- Proofs of the last two involve differential operator D
1. distinct real roots: y = c1 er1 x + . . . + cn ern x

24

3. AMATH
2. repeated real roots (multiplicity k): erx , xerx , . . . , xk−1 erx
3. complex roots (α ± iβ): eαx cos(βx),

eαx sin(βx)

2 & 3. Repeated complex roots:
eαx cos(βx), eαx sin(βx), . . . , xk−1 eαx cos(βx), xk−1 eαx sin(βx)
– Application
∗ Mass spring damp: mx′′ + cx′ + kx = 0
∗ pendulum: s = lθ,

mlθ′′ = −mg sin(θ)

Two models are of the same form y ′′ + b1 y + b0 y = 0
b21
b21
b21

b1 = 0
− 4b0 < 0
− 4b0 = 0
− 4b0 > 0

two complex root
one real repeated root
two real roots

underdamped
critically damped
overdamped

simple harmonic motion
oscillatory with amplitude decaying
not oscillatory
not oscillatory

– Non-homogeneous DE
∗ Undetermined Coefficients
∗ Variation of Parameters
∗ Application
· Forced, undamped motion: resonance and beating
· Forced, damped motion: practical resonance
• Linear Systems of DEs
– definition x′ = P (t) x + f(t),
±

x(t0 ) = x0

coefficient
matrix

– E/U: P (t), f(t) are continuous on an open interval I containing point t0 , then there
exists a unique solution on I.
– Superposition
– Wronskian of x1 , . . . , xn (which are solutions of x′ = P (t)x) is
⎡ x11 (t) . . . xn1 (t) ⎤
⎢
⎥
⎢
⎥
⋱
⋮ ⎥
W (x1 , . . . , xn ) = det(M ) = det [x1 (t) . . . xn (t)] = det ⎢ ⋮
⎢
⎥
⎢x1n (t) . . . xnn (t)⎥
⎣
⎦
dependent, W ≡ 0; independent, W ≠ 0, ∀t ∈ I.
– General Solution of Homogeneous/Non-Homogeneous Linear Systems
⃗′ = P (t)⃗
x
x

(3.2)

Proof of Homogeneous one (responsible for final)
Proof
⃗(t) be any solution on I of (3.2). Let t0 ∈ I, and M (t) be as in the definition
▸ Let x
⃗1 , . . . , x
⃗n are linearly independent on I,
of the Wronskian. Since x
⃗n (t0 )) ≠ 0
det(M (t0 )) = W (⃗
x1 (t0 ), . . . , x
⃗(t0 )
Thus the linear system M (t0 )⃗
c=x
has a unique solution
⎛ c1 ⎞
⎜c ⎟
c⃗ = M −1 (t0 )⃗
x(t0 ) = ⎜ 2 ⎟
⎜⋮⎟
⎝cn ⎠

25

(∗)

3. AMATH
⃗n (t). This is a solution of (3.2) by the Superposition
⃗1 (t)+. . .+cn x
Define y⃗(t) = c1 x
⃗(t0 ). But x
⃗(t) is also a solution
Principle and satisfies the initial condition y⃗(t0 ) = x
of (3.2) satisfying the same IC. By the E/U Theorem we must have
⃗(t) = y⃗(t) = c1 x
⃗1 (t) + . . . + cn x
⃗n (t)
x

– Eigenvalue Method 1
λ1 , λ 2 ∈ R
v1 , v2
λ
v, u
λ1,2 = α ± iβ v1,2 = u ± iw

∀t ∈ I

c1 eλ1 t v1 + c2 eλ2 t v2
c1 eλt v + c2 (eλt u + teλt v)
c1 eαt (cos(βt)u − sin(βt)w) + c2 eαt (sin(βt)u + cos(βt)w)

– solution curves
∗ saddle point: nonzero distinct eigenvalues of opposite sign

∗ Nodes (sink): distinct negative eigenvalues. Origin: improper nodal sink

∗ Nodes (source): distinct positive eigenvalues. Origin: improper nodal source

1

(A − λI)u = v, u is a generalized eigenvector of λ

26

3. AMATH

∗ Repeated positive eigenvalue.
· with two independent eigenvectors. Origin: proper nodal source

· without two independent eigenvectors. Origin: improper nodal source

∗ Repeated negative eigenvalue.
· with two independent eigenvectors. Origin: proper nodal sink (5.3.8)

27

3. AMATH
· without two independent eigenvectors. Origin: improper nodal sink (5.3.9)

∗ Complex conjugate eigenvalues and eigenvectors
· pure imaginary: center
· negative real part: spiral sink
· positive real part: spiral source

– Fundamental Matrix: Φ(t) = [x1 (t) . . . xn (t)], where x1 , . . . , xn ∈ Rn are n
linearly independent solutions of x′ = P (t)x on I.
● Propositions
∗ Every solution x(t) can be written x(t) = Φ(t)c where c ∈ Rn .
∗ invertible
∗ Φ′ (t) = P (t)Φ(t)
● Theorem (Fundamental Matrix Solution)
x′ = P (t)x,

x(t0 ) = x0

unique solution is

x(t) = Φ(t)Φ−1 (t0 )x0 , t ∈ I

– Nonhomogeneous Linear Systems: Variation of Parameters. x′ = P (t)x + f(t)
ˆ
x(t) = xh (t) + xp (t) = Φ(t)c + Φ(t) Φ(t)−1 f(t) dt
• Laplace Transforms

28

3. AMATH
ˆ

∞

– Definitions: F (s) = L {f (t)} =

e−st f (t)dt
0

⎧
⎪
⎪0 t < a
– Unit step: ua (t) = u(t − a) = ⎨
⎪
⎪
⎩1 t ≥ a
– exponential order: ∣f (t)∣ ≤ M ect ,
for t ≥ T

(2)

– Existence of the Laplace Transform (responsible for final): If f is piecewise continuous on t ≥ 0 and of exponential order as t → ∞ with constant c in eq(2), then
L{f (t)} = F (s) exists for s > c.
converge absolutely Ô⇒ converges Ô⇒ exists for s > c
Proof
▸ Since f is piecewise continuous on t ≥ 0, we can find M ≥ 0 such that (2) is
satisfied with T = 0. i.e.
∣f (t)∣ ≤ M ect ,

for t ≥ 0

´∞

´∞

0

0

M ect e−st dt converges if s > c. Thus using a comparison theorem
converges for s > c.
´∞
It follows that 0 f (t)e−st dt converges.

∣f (t)e−st ∣dt

– Gamma function
– Uniqueness of the Inverse Laplace Transform
– Transform of Derivatives L {f ′ (t)} = sL {f (t)} − f (0) = sF (s) − f (0)
– Corollary
L {f (n) (t)} = sn L {f (t)} − sn−1 f (0) − sn−2 f ′ (0) − . . . − f (n−1) (0) = F (s)
– Theorem (Laplace Transform of Integrals) responsible for final
If f (t) is piecewise continuous on t ≥ 0 and is of exponential order as t → ∞ (with
constants c, T, M ) then
ˆ t
1
F (s)
L { f (τ)dτ} = L {f (t)} =
for s > c
s
s
0
equivalently:
L

−1

F (s)
{
}=
s

ˆ

t

f (τ)dτ
0

Proof
´t
▸ Since f is piecewise continuous, on t ≥ 0. g(t) = 0 f (t)dt is continuous on t ≥ 0,
g ′ is piecewise continuous on t ≥ 0.
Further,
ˆ t
ˆ t
ˆ t
M ct
M ct
(e − 1) ≤
e
t≥0
∣g(t)∣ = ∣ f (τ)dτ∣ ≤
∣f (τ)∣dτ ≤
M ecτ dτ =
c
c
0
0
0
So g(t) is of exponential order as t → ∞ and we can apply the Theorem on Laplace
Transform of Derivatives.
L {f (t)} = L {g ′ (t)} = sL {g(t)} − g(0) = sL {g(t)}
ˆ t
1
Ô⇒ L { f (τ)dτ} = L {g(t)} = L {f (t)}
s
0

29

for s > c
for s > c

3. AMATH
– Translation: L {eat f (t)} = F (s − a),

s>a+c

– differentiation of transforms: L {−tf (t)} = F ′ (s)
– convolution: L {f (t) ∗ g(t)} = F (s) ⋅ G(s)
– translation: L {u(t − a)f (t − a)} = e−as F (s)

for s > c

• Appendix

ˆ x
dy
= f (x, y), then yn (x) = y0 +
f (t, yn−1 (t))dt.
– Method of Successive Approximations:
dx
x0
– Existence for Linear Systems. The IVP has a solution on the entire interval I.
x′ = P (t)x + f(t),

x(a) = b

3.3. AMATH 390
3.3.1. Topics
This course provides an introduction to some of the deep connections between mathematics and music; mathematics will be used to provide insights into several important aspects of
music. Topics covered include: modelling the acoustics of string, wind and percussion instruments with 1D and 2D partial differential equations, pitch and harmonics, frequency response
and signal sampling with Fourier transforms, and the advantages and disadvantages of various
scales and tuning systems (Pythagorean and just intonation, equal and well temperament).

3.4. AMATH 391

30

4
PMATH

31

5
CO

来到了我最喜欢的部分之一. . . . . . 其实math 2x9 按照课程编码应该算到basic math里面
的，不过鉴于课程内容，我还是把它们都放在这里。
其实CO是两个学科的组合：Combinatorics and Optimization, 组合和优化。两个是有一
定关联的，但还是分开讲了。就像math229/239/249 是introduction to combinatorics，
co227/250/255 是introduction to optimization。当时一开始我没弄清楚这两个的差别，慢慢
等到上了这些课就明白了，在intro这个部分，两门课的联系不是很大。1

1

收回刚才说的话. . . . . . 今天（2018.10.22）math249刚讲了introduction to graph theory，然后下午co255就
讲 了totally unimodular matrix在bipartite graph等 的 应 用 。 这 个 是 在advanced讲 了 ， 不 知 道250会 不 会
讲. . . . . . anyway，我得补习一下什么是matching and perfect matching了。今天（10.23）又讲了cover。

32

5. CO

5.1. Overview
linear
algebra

co 250
Intro Optim.

math 239/249

co 255 (adv)

Intro Comb.

Intro Optim.

330
comb.
enum

342
intro
graph

430
algebra
enum

442
graph

351
network
flow

331
coding
444
algebra
graph

446
matroid

450
combinatorial
optim.

353
discrete
optim.

367
nonlinear

452
int.
optim.

463
convex
optim.

pmath

pmath

346/347

351

466
cont.
optim.

471
semidefinite
optim.
487
Applied
Cryptography

485
Public-Key
Cryptography

math239/249
co250/255
453
network
design

439
topic:
enum

440
topic:
graph

454
schduling

459
topic:
optim.

33

456
intro
game
theory

Part II.

CS
6. Basic

35

6.1. CS 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

6.1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

6.2. CS 136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

6.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

7. Major

36

7.1. CS 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

7.1.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

7.2. CS 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

7.2.1. Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

7.2.2. Diary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

8. Minor

38

9. Beyond

39

6
Basic

6.1. CS 135
6.1.1. Overview

CS 135
syntax &
semantics

struct

recursion

tree

local

design recipe

list

sort

bst

lambda

generative
recursion

CS history
(will be tested)

Graph Theory
(just joking...)

6.2. CS 136
6.2.1. Overview

Dave’s CS 136 (before Nomair)
Racket→C

pointers

modularization

!arrays!

imperative

efficiency

Model
control
flow

time

memory

35

space

strings

Heap
ADT

linked
list
ADT:
stack
sequence
queue
tree
bst
dictionary

7
Major

先说一下CS的 big three。452（小火车），444（compiler），488（graphics）。然后
还有一门也很有意思442（Principles of Programming Languages）。大二公认比较难的就
是246，241。大三就是350，349。
然后是算法big three（不知道他们怎么叫），341→466(666)→761。三门课越往后越抽
象。但是很多工作其实用不到很多666，761的算法，因为已经超越到一个境界了. . . . . .

7.1. CS 246
7.1.1. Summary
Note that it is NOT guaranteed that all the topics covered in class are listed here.
• linux shell (bash)
– commands

– inheritance

– RAII

– Design patterns

– piml

– globbing

∗ UML

– script

∗ factory
∗ observer

• C++

∗ singeleton

– basic

∗ tempalte

– stream

∗ decorator

– reference

∗ visitor

– overloading

∗ iterator

– include guard

∗ bridge

– Classes

• STL

∗ Big 5

– vector

∗ MIL
∗ copy-andswap
∗ rvalue
lvalue

and

∗ linked lists
∗ encapsulations

– MVC
• Other
– makefile
– debugging
∗ gdb
∗ valgrind
– exception
– smart ptrs
– vtable, vptr
– cast

– stack

– forward
tion

– queue

– template

– set

– STL algorithms

– maps

declara-

∗ for each
∗ transform

• idiom
– NVI

36

– lambda

7. Major

7.2. CS 251
7.2.1. Topics

A mess
7.2.2. Diary
Midterm
现在终于弄懂了fsm是怎么表示的了。1 明天下午4点考，现在真的很慌，因为真的学的
不是很认真，上课也没好好听，现在就在临时抱佛脚。但愿今晚能多复习一点，虽然现在已
经快12点了. . . . . .
突然回想到上次考试前几乎什么都不会的还是phys111，那会还好应付，而且还是期
末。大概是下午4点考，然后从晚上睡不着，大概从凌晨4点一直看到考试前。现在没那么好
应付了，然后我现在还在 typesetting LATEX. . . . . .

2018.10.22

Midterm
转点了，先睡了，早起复习，再git push 一下.
After mid, 感觉凉透了. . . . . . 考试有很多东西感觉没复习到位，就像在学习新知识一
样。回忆起来上次，考前一小时还在复习的是stat231 mid1，不过那个好歹我每节课都去听
了，虽然不是每节课都完全能接受，相比251已经是一个巨大的比例了。上课能大大减少我
在课外花的时间。学长的建议：

2018.10.23

2

前期比较简单，但老师讲的不是很清楚，slide很混乱，后面有点加大难度。但
是这门课没什么逻辑可言，只要把该背的都背了，背好简单的逻辑，上课好好听
课，然后不懂的就去问prof，还是可以熬过这门课的。课还是挺水的。
真的每个人都说这门课水，从某种角度来说确实是这样的。所以还是慢慢学吧，更多的
等再学一段时间再说吧。我其实更希望整个slide能像cs135/136那样明确，有很明确的划分，
有很明显的重点，和考点。ppt给我的感觉就是一个页面一个标题，然后想到啥写啥，该明
确的地方可能一句话带过了。可能是这门课的course coordinator对这门课的维护不是很到
位，只是想着把改覆盖的东西一股脑的丢上去就完了。但愿这门课几年后能有所改变，不过
现在这个教授已经教了好像有好几年了，slide也是一直没有变化。但是她人挺好的，经常会
在课上复习，也会经常回答同学的问题。
After Mid
成绩出了，果然炸了。前几个学期是比这学期期中简单不少，唉. . . . . . 现在又在讲single
cycle啥的，越来越迷了，slide也越来越乱了. . . . . .

1
2

如果你早就会了，不要嘲笑学长，学长好菜. . . . . .
cs246(e)会稍微讲一讲

37

2018.10.25

8
Minor

38

9
Beyond

39

Part III.

MUSIC
10. Music Theory
10.1. MUSIC 111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Music Ensemble

41
41
42

10
Music Theory

这个地方是我最喜欢的部分之一。乐理是作曲的基础，也是很多学习乐器中的必修课。
难点肯定是有的，但是学起来不会很枯燥，会很有趣。相比较pmath的课肯定有趣不少，也
不会枯燥的，当然这样比较其实没有太大价值. . . . . .

10.1. MUSIC 111
这门课的title叫做Fundamentals of Music Theory，就是rudiment，非常基本的乐理知
识，是给那些几乎没有基础的人上的，因为它真的是从最基本的音符，五线谱到最后的3和
弦和一些调式为止。可能对于那些从来没学过乐理的人信息量有点大，不过紧跟步伐，认真
做作业还是很好理解内容的。否则的话，我就不建议学乐理了. . . . . .

41

11
Music Ensemble

顾名思义，这个课就是一个小型乐团课，一门课排练一学期，然后最后汇报演出。课号
为 MUSIC 116 117 216 217 316 317。上完一门如果还想上下一门就接着上。如果你不想要
学分，不enroll这门课也没有关系。接下来为了让整体格式整齐，我先用FAQ的格式来回答
一些问题。
乐团有哪些选择呢？
Answer: 根据官网所列的，大概有以下几种
• Chamber Choir 室内合唱团
• Chapel Choir 教堂唱诗班
• University Choir 大学合唱团，全称叫 UW Choir
• Vocal Techniques 声乐技巧
• Instrumental Chamber Ensembles 室内乐团
• orchestra@uwaterlo 管弦乐团
• Jazz Ensemble 爵士乐队
• World Music Ensemble: Balinese Gamelan 世界音乐乐团：甘美兰1
如何才能进入这些乐团呢?
可以去官网的每个乐团的网站，里面有详尽的介绍。一般来说都是需要试音的
（audition）。但是对于有些乐团，如果你之前在这个乐团里，下学期如果你还想
来，只需要跟instructor发邮件就好，表明你的意愿。
听说可以拿学分！！！？？
是的，这门课有0.25学分，而且不用交学费（曲谱的费用除外，当然不会超过100刀
的）。而且这门课是直接可以加在正常的course load 里面的，有些时候加不上去可以
找advisor帮忙调整。当然前提是你enroll了乐团课，然后也过了audition，然后缺勤太
多次，最后会在成绩单上有0.25学分，听来是不是很酷！！
那 instructor是
是谁 呢 ？
每个乐团的我会详尽在后面每个chapter说明，不过他们都是非常有经验的音乐家，一
定会让你好好享受音乐的。
排练会不会占用太多时间呢？
如果课程不是非常满的话是不会的，除非你每天像学长一样一堆due，赶mid、quiz等。
一般来说都是每周排练一次，大概3-4小时，具体的时间我会在接下来的章节详尽说
明。
1

印度尼西亚历史最悠久的一种民族音乐

42

11. Music Ensemble
一般在哪里排练呢？乐器会不会很水？
不会的. . . . . . 这是我们学校的音乐系组织的乐团，在Renison旁边有个专门的附属学校
就是它—Conrad Grebel College. 一般在那里排练。钢琴都是大三角雅马哈，虽然不是
施坦威，但是也是非常不错的琴，毕竟很多教授都是从Laurier直接过来的，配置当然
不能低。当然像小提琴这种还是自己带啦。

43

Part IV.

OTHER
12. Science

45

13. Arts

46

12
Science

45

13
Arts

46



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