成大數學系大學部課程總覽

編號 課程碼 屬性碼 課程名稱 Course Name

1 C111210 Math1161 線性代數(一) Linear Algebra I

2 C111300 Math1131 數學導論 Introductory Mathematics

3 C111400 Math1141 計算機概論與程式語言 Introduction to Computer and Programming Language

4 C112100 Math1151 解析幾何與矩陣 Analytic Geometry and Matrix

5 C112210 Math1111 微積分(一) Calculus I

6 C112220 Math1121 微積分(二) Calculus II

7 C117910 Math1101 服務學習(一) Service Learning I

8 C117920 Math1102 服務學習(二) Service Learning II

9 C111220 Math2101 線性代數(二) Linear Algebra II

10 C117930 Math2151 服務學習(三) Service Learning III

11 C120500 Math2131 微分方程導論 Introduction to Differential Equations

12 C121110 Math2111 高等微積分(一) Advanced Calculus I

13 C121120 Math2121 高等微積分(二) Advanced Calculus I

14 C121310 Math2141 代數學(一) Algebra I

15 C121400 Math3707 科學計算導論 Introduction to Scientific Computing

16 C122500 Math2214 科學計算軟體 Introduction to Scientific Computing Software

17 C144800 Math5201 應用分析 Applied Analysis

18 C120600 Math3201 機率導論 Introduction to Probability

19 C122100 Math1152 數論導論 Introduction to Number Theory

20 C122000 Math3601 統計導論 Introduction to Statistics

21 C137000 Math3602 統計科學 Statistical Science

22 C147400 Math3603 統計推論 Statistical Inference

23 C126600 Math3401 離散數學 Discrete Mathematics

24 C131800 Math5404 拓樸學 Topology

25 C133520 Math3203 代數學(二) Algebra II

26 C133710 Math5400 幾何學(一) Geometry I

27 C133800 Math5202 複變數函數論 Complex Analysis

28 C131121 Math5202 複變數函數論(二) Complex Analysis

29 C133900 Math5800 數值分析(一） Introduction to Numerical Analysis

30 C134300 Math5205 動態系統 Dynamical Systems

31 C135000 Math5001 群表現導論 Representation Theory of Finite Groups

32 C135100 Math5200 初等分析 Undergraduate Analysis

33 C135300 Math3702 金融數學導論 Introduction to Financial Mathematics

34 C135500 Math3703 數學規劃導論 Mathematical Programming

35 C135600 Math3704 數學建模 Mathematical Modeling

36 C135700 Math3705 凸分析導論 Convex Analysis

37 C135800 Math3305 通過問題學數學 Mathematics - Learning through Problems

38 C136300 Math3304 向量分析 Vector Analysis

39 C146000 Math5805 金融市場與金融數學導論 Introduction to Financial Markets and Financial Mathematics
40 C137100 Math5810 金融體系、金融市場與金融數學導論 Introduction to Financial System, Financial Markets and Financial Mathematics

41 C122600 Math4603 線性模式 Linear Model

42 C130121 Math5407 幾何學(二) Geometry II

43 C142300 Math5405 非歐幾何 Non-Euclidean Geometry

44 C143700 Math5206 偏微分方程導論 Introduction to Partial Differential Equations

45 C143800 Math4601 隨機過程 Stochastic Processes

46 C144901 Math5402 流形上的微積分 Calculus on Manifolds

47 C145300 Math5801 應用數學方法 Applied Mathematics Mathods

48 C145500 Math5000 伽羅瓦理論入門 Introduction to Galois Theory

49 C145600 Math5002 計算代數 Computational Algebra

50 C145700 Math5403 流形導論 Introduction to Manifolds

51 C145800 Math4602 數值微分方程導論 Introduction to Numerical Differential Equations

52 C146100 Math5204 傅立葉分析與應用 Fourier Analysis and Applications

53 C146200 Math4602 應用機率 Applied Probability

54 C146600 Math4405 矩陣群 Matrix Groups

55 C136100 Math3404 賽局論 Game Theory

56 C145100 Math5803 語音訊號處理 Speech Signal Processing

57 C137500 Math3708 數據科學裡的矩陣方法 Matrix Methods in Data Science

課程名稱 線性代數(一) Linear Algebra I

課程簡介這門課以線性方程及矩陣為開端，再逐一引進線性代數的基本概念，包括向量空間、線性變換、基底與維度、行列式、特徵值與特徵向量以及對角化。 In this course, we begin by discussing linear equations and matrices. We then proceed to introduce the basic concepts of linear algebra. Topics include vector spaces, linear transformations, bases and dimension, determinants, eigenvalues and eigenvectors, and diagonalizability.

課程名稱		數學導論	Introductory Mathematics
課程簡介		本課程的目的是要培養學生以數學方法來思考，並且能了解並且創造數學證明。此課程將幫助學生發展對嚴謹的證明的分析及寫作能力。	This course provides a fundamental foundation for working with advanced mathematics. In order to achieve advanced mathematics course, precise language and methodology must be used.

課程名稱		計算機概論與程式語言	Introduction to Computer and Programming Language
課程簡介		本課程目的在簡介計算機系統簡介與基本的程式語言撰寫訓練。在計算機概論部分，課程涵蓋：二進位系統、資料表示方式、邏輯閥、硬體架構、網路以及作業系統等等。在程式語言撰寫部分，課程涵蓋：程式語言的基本介紹、偵錯技巧、解題過程以及演算法的應用。課程目的在訓練學生，使其能夠撰寫程式解決簡單的問題。	This course is provided for students with little or no prior computer and programming experience. It aims to provide students with an understanding of computer science and programming. To give an introduction to computer, topics including binary values and number systems, data representation, logic gates, hardware organization, the Internet, and operating systems are covered. In computer programming, the course covers the basics of a programming language, simple debugging process, problem solving processes and implementation of algorithms. It aims to help students to establish their ability to write small programs that allow them to accomplish useful goals.

課程名稱		解析幾何與矩陣	Analytic Geometry and Matrix
課程簡介		3-4週的解析幾何包含：圓錐曲線、極座標、參數方程、向量、內積、外積、行列式、多變數函數、等值集合、切平面 14-15週的線性代數包含：行列式、反矩陣、聯立方程組、Cramer’s公式、特徵向量與特徵值、特徵方程式、對角化、向量空間、子空間、基底、線性轉換、規範形式、二次形式、應用 Reference for Analytic Geometry: Chapter 10, 13, 15 of the Calculus: One and Several Variables, 10th Edition by Satunino L. Salas, Einar Hille and Garret J. Etgen Reference for Linear Algebra: Chapter 1, 2, 3, 5, 8 of the Applied Linear Algebra by Peter J. Olver and Chehrzad Shakiban	3-4 weeks on Analytic Geometry: Conic Sections, Polar Coordinates, Parametric Equations, Vectors in Three-Dimensional Space, The Dot Product, The Cross Product, Determinants, Functions of Several Variables, Level Sets, Tangent Planes 14-15 weeks on Linear Algebra: Determinants, Invertible Matrices and Inverses, Systems of Linear Equations, Cramer's Rule, Eigenvectors and Eigenvalues, The Characteristic equation, Diagonalization, The Vector Spaces, Subspaces, Bases, Linear Transformations, Canonical forms, Quadratic forms, Applications. Reference for Analytic Geometry: Chapter 10, 13, 15 of the Calculus: One and Several Variables, 10th Edition by Satunino L. Salas, Einar Hille and Garret J. Etgen Reference for Linear Algebra: Chapter 1, 2, 3, 5, 8 of the Applied Linear Algebra by Peter J. Olver and Chehrzad Shakiban

課程名稱		微積分(一)	Calculus I
課程簡介		本課程以極限的觀念為基礎，建立微分與積分的運算並以之探討實函數之局部性質與整體行徑。	Based on the concept of limit, we discuss both differential calculus and integral calculus. These will be used as tools to discuss the local and global behavior of real-valued functions.

課程名稱		微積分(二)	Calculus II
課程簡介		本課程以微積分(一)的觀念為基礎，我們將討論在多維空間中函數的微分和積分。我們亦討論向量值函數和多變量的函數，並且學習微積分基本定理的推廣：Green-Stoke定理和Divergence定理以及其應用。	Based on the previous course , we discuss both differential calculus and integral calculus in multidimensional real spaces. We consider vector-valued functions and functions of several variables, and study a generalization of the fundamental theorem of calculus to multi-variables: Green-Stoke Theorem and Divergence Theorem, with their applications.

課程名稱		線性代數(二)	Linear Algebra II
課程簡介		這門課介紹矩陣的對角化，內積空間的一些基本性質和一些重要的算子。最後，我們討論Jordan標準形，極小多項式和有理標準形。	In this course, we introduce Diagonalization of matrices, some basic properties of Inner Product Spaces, and some important operators. Finally we discuss Jordan Canonical Forms, Minimal Polynomial, and Rational Canonical Forms.

課程名稱		微分方程導論	Introduction to Differential Equations
課程簡介		微分方程是用來描述科學和工程現象的重要工具。本課程係介紹線性常微分方程式，主要包含：微分方程解的存在唯一性，解微分方程的技巧，方程解的定性分析，微分方程的近似解與數值解。	Differential equations are important tools to understand phenomena in science and engineer. This course is aimed to introduce linear ordinary differential equations (ODEs). The topics include: existence and uniqueness of solutions of ODEs, techniques of solving ODEs, qualitative analysis of ODEs, approximated solutions and numerical solutions of ODEs.

課程名稱		高等微積分(一)	Advanced Calculus I
課程簡介		這門課介紹微分、積分及函數性質，重點為定理的證明。我們研究的實數系和基本的拓撲結構，然後討論序列和級數。接下來，我們討論函數的連續性，微分、積分。最後，我們將談論函數所形成的序列和級數。	Real number system, Euclidean n-space Point set topology: open and closed sets, boundary of a set, sequence and series Compact and connected sets: Heine-Borel and Bolzano-Weierstrass theorems, path-connected and connected sets Continuous mappings: boundedness of continuous functions on compact sets, intermediate-value theorem, uniform continuity Differentiation: matrix representation, Taylor’s theorem, maxima and minima Uniform convergence of sequence of functions: pointwise and uniform convergences, integration and differentiation of series, Arzela-Ascoli theorem, fixed point theorem, Stone-Weierstrass theorem, Dirichlet and Able test

課程名稱		高等微積分(二)	Advanced Calculus II
課程簡介		這門課介紹微分、積分及函數性質，重點為定理的證明。我們研究的實數系和基本的拓撲結構，然後討論序列和級數。接下來，我們討論函數的連續性，微分、積分。最後，我們將談論函數所形成的序列和級數。	Uniform convergence of sequence of functions: pointwise and uniform convergences, integration and differentiation of series, Arzela-Ascoli theorem, fixed point theorem, Stone-Weierstrass theorem, Dirichlet and Able test Differentiable mappings, matrix representation, conditions for differentiability, Taylor's theorem and higher derivatives, maxima and minima Contraction principle, inverse function theorem, implicit function theorem, rank theorem. Lagrange multipliers, existence theorem for ODE Integration: integrable functions, volume, measure zero, Lebesgue's theorem, improper integral, Fubini's theorem, change of variables, polar, spherical, cylindrical coordinates, interchanges of limiting operations Fourier analysis: Fourier series, inner product spaces, orthogonal families, completeness and convergence theorems, functions of bounded variation, Fejer's theorem

課程名稱		代數學(一)(二)	Algebra I and II
課程簡介		代數學(一)及(二)延續了線性代數課程，但是其題材更為抽象，也有更多證明的訓練。在此一課程中，會介紹群、環、體等結構，發展研究其結構的方法，並且將其應用在不同的領域中。此一課程會持續地帶領學生深入了解數學，也增強學生在以數學符號及口頭來溝通數學，讓學生能優游自在地閱讀、了解數學，進一步培養對抽象數學的鑑賞能力。	Algebra (1) and (2) is a natural continuation of Linear Algebra, but the material is much more proof-driven and abstract. During the year, the structures groups, rings, and fields will be introduced, and the method of studying them will be developed, and applications to other areas will be seen. The general objective of this course is to continue providing students with a deeper understanding and working knowledge of mathematics, strengthening the analytic skills, increasing the ability to communicate mathematics symbolically and orally, making students comfortable with reading and understanding mathematics on their own, and continuing to develop the appreciation for abstract mathematics.

課程名稱		科學計算導論	Introduction to Scientific Computing
課程簡介		本課程將介紹一些基礎科學計算工具，包括演算法與計算方法。對於這堂課我們的學習觀點就是”作中學” 。因此，除了老師講課之外，我們會提供有關現實問題的案例研究，引導學生通過實際操作來解決問題。目前暫定課程內容如下(如有需要可能會有變動): 1.函數計算，錯誤分析和複雜度計算 2.密集/稀疏矩陣計算：矩陣分解 3.特徵值問題 4.求解線性和非線性方程的方法 5.插補（回歸） 6.數值優化	This course is to introduce some fundamental scientific computing tools including algorithms and computing techniques. Learning by doing is the viewpoint we will keep in this course. Thus, in addition to lectures, we will provide cases studies on real world problems and guide students to solve the problems by hand-on activity. The tentative course contents are listed as follows, but are subject to change if required. 1.Functional evaluation, error analysis, and computational complexity 2.Dense/Sparse matrix computation: matrix factorizations 3.Eigenvalue problems 4.Methods for solving linear and nonlinear equations 5.Interpolation(regression) 6.Numerical optimization

課程名稱		科學計算軟體	Introduction to Scientific Computing Software
課程簡介		這門課我們將介紹用於科學計算之軟體或程式語言(Matlab或類似之科學計算軟體)，使學生熟悉程式基本之設計，學習使用科學計算軟體完成數學及工程問題的分析與運算，以便作為未來使用科學計算問題之基礎。	This course provides an introduction to the use of Matlab or similar software to solve problems arising in multiple science and engineering domains. The course covers application of mathematical judgment, programming architecture, and flow control in solving scientific problems. The aim of this course is to equip students with a competence to apply Matlab or similar software as a tool for problem solving. The course is in particular recommended to students who are interested in doing scientific computing.

課程名稱		應用分析	Applied Analysis
課程簡介		這是一門讓學生學習那些具有實際應用的分析入門課程。課程的內容包括：度量與賦範空間、連續函數、收縮映射定理、拓樸空間、巴拿赫空間、希耳伯特空間、傅立葉級數、(如果時間允許)希耳伯特空間上的有界算子。	The aim of this course is to supply an introduction for students to those parts of analysis that are most useful in applications. The following material will be covered in this one semester course: Metric and Normed Spaces, Continuous Functions, The Contraction Mapping Theorem, Topological Spaces, Banach Spaces, Hilbert Spaces, Fourier Series and (if time permitted) Bounded Linear Operators on Hilbert Spaces.

課程名稱		機率導論	Introduction to Probability
課程簡介		集合論與機率之定義，離散與連續之隨機變數，聯合機率變數與分配，動差生成函數，特徵函數，極限理論。	Definition of Probability, Random Variable, Joint Random Variable and it's Distribution, Moment Generating Function, Characteristic Function, Limiting Theory.

課程名稱		數論導論	Number Theory
課程簡介		數論入門，模組化算術，中國留數定理，費瑪小定理...等等。	Introduction to number theory, modular arithmetic, Chinese remainder theorem, Fermat's little theorem and etc.

課程名稱		統計導論	Introduction to Statistics
課程簡介		敘述統計，樣本分配，點估計，區間估計，假設檢定，適合度檢定。	Describe Statistics, Sampling Distribution, Point Estimation, Interval Estimation, Testing Hypothesis, Goodness of fit test.

課程名稱		統計科學	Statistical Science
課程簡介		這門課的開設是適合沒有修過統計相關課程或是想要在學習更進階的統計課程前而複習統計基本知識的大學生。本課程將會介紹統計方法的基礎概念和如何使用統計軟體 R來做資料分析。	This course is designed for undergraduate students who either have no prior statistics courses or who wish to review the basic statistics before taking additional advanced statistics courses. This course will present the fundamental concepts in statistics and give an introduction to R statistical software for preliminary data analysis.

課程名稱		統計推論	Statistical Inference
課程簡介		本門課程是設計給高年級數學系的學生或主修統計且具有清晰邏輯思考數學背景的學生。這門課透過扎實的訓練學習，提升學生關於機率與數理統計的介紹與理解。	This course is designed for advanced seniors in mathematics or statistics with strong mathematical backgrounds. This course is aimed to provide a solid and well-balanced introduction to probability and mathematical statistics.

課程名稱		離散數學	Discrete Mathematics
課程簡介		排列，組合，圖論，遞歸算法...等等。	Combinatorics, graph theory, recursive algorithms and etc.

課程名稱		拓樸學	Topology
課程簡介		拓樸學研究幾何物體(拓樸空間)和它的分類問題。它探討物體在連續形變下不變的性質，譬如：正方形和圓形在拓樸上是一樣的。這門課將從拓樸空間和連續函數的定義出發，接著引進連通性、緊緻性、緊緻化、商空間、正規空間等基本概念。我們也探討Urysohn 引理和Tychonoff 定理，以及淺談基本群及覆疊空間。	In topology, we study geometric objects (topological spaces) and their classification. We are interested in properties of objects that are preserved when one object is continuously deformed into another. For instance, a square and a circle are topologically the same. In this course, we will begin by defining what topological spaces and continuous functions are. We will then discuss connectedness, compactness, compactification, quotient spaces, normal spaces, Urysohn’s Lemma, and Tychonoff’s Theorem, and give an elementary introduction to fundamental groups and covering spaces.

課程名稱		幾何學(一)	Geometry I
課程簡介		這門課的目的是研究曲線和曲面的幾何性質。我們將討論曲線的曲率和撓率、Frenet–Serret 公式、高斯映射、高斯曲率與均曲率、曲面的基本形式、測地線、Gauss–Bonnet 定理等課題。	This course is designed to study the geometric properties of curves and surfaces. We will talk about curvatures and torsions of curves, Frenet–Serret formulas, Gauss map, Gaussian and mean curvatures, fundamental forms of surfaces, geodesics, Gauss–Bonnet Theorem, etc.

課程名稱		複變數函數論	Complex Analysis
課程簡介		這門課主要是介紹基本的複變數函數的理論。主要的內容包括複數系、解析函數、初等函數、複數積分、解析函數的數列表示、留數理論、保角映射以及 Fourier 和 Laplace 轉換。除了基礎理論，我們也將簡單介紹一些在物理或工程方面的應用。	This course provides the mathematical foundations needed for understanding and working with the modern analysis. The main topics are complex number system, analytic functions, integrations, conformal mappings and some integral transformations. Some applications in physics and engineering will also be included.

課程名稱		複變數函數論(二)	Complex Analysis II
課程簡介		解析函數複習、曲線積分與解析函數、單連通區域、曲線積分技巧、黎曼映射定理、極大值定理、調和函數、解析延拓、應用。	Review of Analytic Functions, Line Integrals and Analytic Functions, Simply Connected Domains, Further Contour Integral Techniques, The Riemann Mapping Theorem, Maximum-Modulus Theorems for Unbounded Domains, Harmonic Functions, Analytic Continuation; Applications to Other Areas of Mathematics.

課程名稱		數值分析(一)	Introduction to Numerical Analysis
課程簡介		這門課為數值分析的入門課程，主要在介紹一些數值分析的基本概念，主要內容包括：浮點數系統、精度與誤差、一維非線性函數勘根數值方法與收斂性分析、函數的數值逼近與插值函數、數值微分與數值積分、數值微分方程簡介以及數值穩定性分析等。除了理論分析的教導之外，本課程也強調通過基本的科學計算軟體，如：Matlab 等，進行數值實驗與習題操作。	This course provides an introduction to numerical analysis. The course covers elementary concepts of numerical analysis, including: system of floating numbers, precision and error analysis, root finding methods for one dimensional nonlinear functions and convergence analysis, approximation to funtions and data and interpolations, numerical differentialtion and integration, elementary stability analysis for numerical differential equations. In addition, students will be supported to use scientific computing softwares, e.g., MatLab, etc., to their assignments in practice.

課程名稱		動態系統	Dynamical Systems
課程簡介		大自然的現象往往隨時間充滿複雜的變化，且對應的微分方程也是非線性的。本課程係非線性常微分方程的定性分析，主要包含：線性系統的穩定性分析，非線性系統解的局部與廣域分析，分歧理論與混沌。	Most natural phenomena involve complicated changes in time, and the corresponding differential equations are nonlinear. This course is aimed to the qualitative study of nonlinear ordinary differential equations (ODEs). The topics include: stability analysis of linear systems, local and global theory of solutions of nonlinear systems, bifurcation theory and chaos.

課程名稱		群表現導論	Representation Theory of Finite Groups
課程簡介		本課程為表現理論(包含群和李代數)的入門課程。主要內容為有限群表現理論的基本定義及特性，表現和特徵標的關係。	The goal of this course is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.

課程名稱		初等分析	UNDERGRADUATE ANALYSIS
課程簡介		這門課的主題為metric spaces and normed spaces，基本上可視為高等微積分的延伸。我們將研習一些重要的性質如 boundedness、completeness及compactness。	This course is about metric spaces and normed spaces. It is an extension of the advanced calculus course. We will learn the properties like boundedness, completeness, and compactness in general spaces.

課程名稱		金融數學導論	Introduction to Financial Mathematics
課程簡介		金融市場簡介：股票市場，債券市場，貨幣市場，商品市場，期貨，期權。基本的機率理論，常態分佈，一維的布朗運動，隨機積分，套利，對沖，Black-Scholes模型，一維熱傳導方程，衍生性金融商品。	Introduction to financial markets: stock markets, bond markets, currency markets, commodity markets, Futures, Options. Basic Probability Theory, Normal Distributions, one-dimensional Brownian motion, Stochastic Integration, Arbitrage, Hedge, the Black-Scholes model, one-dimensional Heat Equation, Financial Derivatives.

課程名稱		數學規劃導論	Mathematical Programming
課程簡介		這門課主要介紹線性規劃與無約束條件的最佳化問題，在線性規劃以極端點、線性規劃基本定理、Simplex方法、對偶問題、強對偶定理及對偶問題的最佳解為題材。此外，亦講授無約束問題的解答之基本性質及演算法。	This curriculum mainly introduce the Linear programming problems and unconstrained optimization problem. On the part of the Linear programming problems, we will use the extreme points, Fundamental Theorem of Linear programming problems, Simplex method, dual problem, strong duality theorem and the optimal solution of dual problem as subject. In addition, it will include the basic properties of the solution of unconstrained problem and its algorithms.

課程名稱		數學建模	Mathematical Modeling
課程簡介		本課程介紹不同類別的數學模型，如線性－非線性、確定性－統計性、靜態－動態、離散－連續、演繹－經驗等。並透過物理科學、生命科學、地球科學以及社會科學等不同領域的例子來協助學生以數學語言及觀念來表達及分析生活中事物，並透過所建立的模型進行比對。	Several types of mathematical models are introduced in this course, for example, linear-nonlinear, deterministic-stochastic, static-dynamics, discrete-continuous, deductive-inductive. Models in different fields such as physics, biology, earth science or social science, are illustrated to train students that how to use mathematical languages to describe the nature and validate the models with experimental results.

課程名稱		凸分析導論	Convex Analysis
課程簡介		本課程主要介紹在Rn上的凸集合、Separation Theorems、Affine Geometry、 Supporting Hyperplanes、凸函數、Subgradients、凸函數的微分性質、凸規劃問題、Lagrangian Duality、二次規劃等。	This curriculum mainly introduce the convex set on the Rn, Separation Theorems, Affine Geometry, Supporting Hyperplanes, Convex function, Subgradients, Differentiable Convex Functions, Convex programming problems, Lagrangian Duality and Quadratic programming problems.

課程名稱		通過問題學數學	Mathematics - Learning through Problems
課程簡介		每個大二數學系學生一定得修一年高微，高微常常被學生視為最具挑戰性甚至是最害怕的一科必修課。而開這門課的主要目的是藉由有系統的解問題來引進相關數學的核心概念，使學生更能掌握這門學問，減輕一些不安以及幫助學生考研究所。	Every undergraduate mathematics program requires one-year of Advanced Calculus. Often, students consider this course to be the most challenging or even intimidating. The goal of this course is to alleviate those concerns by systematically solving the problems related to the core concepts of analysis. And we also hope that this course can help them pass the entrance exam.

課程名稱		向量分析	Vector Analysis
課程簡介		向量分析源起於電磁學，發展至今已被廣泛利用於自然科學與應用科技上，例如流體力學、向量可說是力學的自然語言。這門課的內容涵蓋向量分析的基本知識與應用，以多變數的微積分為重點，特別介紹梯度，散度與旋度的概念以及三個重要的積分定理：Green定理，高斯散度定理及Stokes定理。	This course focus on the multi-variable calculus. In particular, the concept of the concept of Div, Grad, Curl, and three great integration theorems (Green’s theorem, the divergence theorem, and Stokes’s theorem) will be well introduced.

課程名稱		金融市場與金融數學導論	Introduction to Financial Markets and Financial Mathematics
課程簡介		在這個課程中，我們將介紹金融市場和一些相關的概念(債券市場，到期收益率或殖利率，股票市場，期貨市場，遠期外匯，掉期，外匯交換及利率)，美國聯邦儲備系統(美國中央銀行)和它的貨幣政策(通過美聯儲的職能的討論)，衍生性金融商品，期權(看漲期權和看跌期權)和期權的數學理論定價(Wiener過程或布朗運動，隨機積分，Black-Scoles模型和熱方程)。我們將通過對2007-2008年的全球金融危機及其後續發展的討論整合金融的各個面向。歡迎主修金融、經濟、數學的學生選修。	In this course, we introduce Financial Markets and some related Concepts (Bond Markets, Yield to Maturity, Stocks Markets, Futures Markets, Forward, Swap, Foreign Exchange and Interest Rates), the Federal Reserve System (the U.S. Central Bank) and its Monetary Policy(through discussions of the functions of Fed), Financial derivatives, Options (Calls and Puts) and the mathematical theory of Options Pricing (Wiener Process or Brownian Motion, Stochastic Integral, Black-Scoles Model, and Heat equation). We will integrate several aspects of Finance through discussions on the Global Financial Crisis of 2007-2008 and its succeeding developments. Students majored in Finance, Economics, or Mathematics are welcome.

課程名稱		金融體系、金融市場與金融數學導論	Introduction to Financial System, Financial Markets and Financial Mathematics
課程簡介		在這個課程中，我們將討論(國際)金融體系、金融市場與相關的數學。因此我們討論Bretton Woods系統與國際貨幣基金(IMF)、國際收支、(幾個關鍵的)中央銀行(例如聯邦儲備系統、歐洲央行、日本央行、中國人民銀行)、銀行系統、外匯交易、貨幣政策對外匯匯率的影響、債券市場、殖利率曲線、股票市場、期貨市場、遠期外匯、掉期、信用違約掉期(CDS)、期權(看漲期權、看跌期權)以及金融商品(債券、外匯匯率、遠期外匯、期權)的定價理論。	In this course, we will discuss(International) Financial System, Financial Markets and the mathematics related. Thus we will discuss the Bretton Woods System and the International Monetary Fund(IMF), Balance of Payments, (certain key) Central Banks(such as Federal Reserve System, European Central Bank, Bank of Japan, and People's Bank of China), Banking System, Foreign Exchange, Influence of Monetary Policy on Currency Exchange Rate, Bond Markets, Yield Curve, Stock Markets, Futures Markets, Forward, Swaps, Credit Default Swap(CDS), Options(Call and Put), and the Pricing Theories for Financial Assets(Bonds, Currency Exchange Rates, Forward, Call option and Put Option).

課程名稱		線性模式	Linear Model
課程簡介		代數矩陣，多變量常態分配，線性迴歸，變異數分析，共軛變異數分析。	Matrix Algebra, Multi-normal distribution, Linear regression, Analysis of Variance, Analysis of Covariance.

課程名稱		幾何學(二)	Geometry II
課程簡介		這門課是幾何學(一) 的延續。我們首先複習幾何學(一) 對 intrinsice geometry of surfaces 的討論，包括平行移動、測地線、Gauss-Bonnet 定理、指數映射。第二部分是專題式討論，可能觸及的題材有曲面的大域性質、抽象曲面、非歐幾何、微分形式在曲面論的應用、極小曲面、均曲率曲面等等。	This course is a continuation of Geometry I. First we review the study of intrinsic geometry of surfaces in Geometry I: parallel transport, geodesics, Gauss-Bonnet Theorem and exponential maps. In the second part we discuss various topics such as the global geometry of surfaces, abstract surfaces, non-Euclidean geometry, differential forms, minimal surfaces, constant mean curvature surfaces, etc.

課程名稱		非歐幾何	Non-Euclidean Geometry
課程簡介		淺談二維(實)雙曲幾何、龐加萊圓盤模型及上半平面模型。	Elementary introduction to two (real) dimensional hyperbolic geometry, the Poincare disk model, and the upper half plane model.

課程名稱		偏微分方程導論	Introduction to Partial Differential Equations
課程簡介		這門課主要是介紹一階偏微分方程及二階線性偏微分方程。我們將學習一些解偏微分方程的古典方法，如特徵線方法、分離變數法、傅立葉轉換法、拉普拉斯轉換法、格林函數方法。	This course provides an introduction to the first-order PDEs and the second-order linear PDEs. We will study the classical approach to solve PDEs, such as the method of characteristics, the separation of variables technique, Fourier transform, Laplace transform, and Green's function methods.

課程名稱		隨機過程	Stochastic Processes
課程簡介		伯努力定理，馬可夫鏈，普松過程，再生過程。	Bernoulli Processes, Mankov chains, Poisson Processes, Renewal Processes.

課程名稱		流形上的微積分	Calculus on Manifolds
課程簡介		這是一門討論Stokes 定理的微分幾何課程。課程的內容包括：反函數定理、隱函數定理、可積函數、Fubini 定理、1 的分割、變數變換、向量場和型式、鏈鎖上的積分、微積分基本定理、流形、流形上的Stokes定理。	This is a differential geometry course centered about Stokes’ Theorem, sometimes called the fundamental theorem of multivariate calculus. Topics include Inverse Function Theorem, Implicit Function Theorem, Integrable Functions, Fubini’s Theorem, Partitions of Unity, Change of Variable, Fields and Forms, Integration on Chains, The Fundamental Theorem of Calculus, Manifolds, Stokes’ Theorem on Manifolds.

課程名稱		應用數學方法	Applied Mathematics Mathods
課程簡介		本課程將介紹在不同的應用數學領域中所需要的基本方法。如尺度分析、因次分析、富氏分析、最佳化方法、微擾法、最速陡降法、牛頓法、共軛梯度法等等。本課程將帶領學生學習從應用層面切入去了解問題並採取適當的解決方案。	Basic methods in applied mathematics are introduced, for example, scale analysis, dimensional analysis, Fourier analysis, perturbation method, steepest descent method, Newton method, conjugate gradient method, etc.. Students will learn how to solve problem in the applied mathematical point of view and take a suitable and reasonable solution.

課程名稱		伽羅瓦理論入門	Introduction to Galois Theory
課程簡介		多項式根，伽羅瓦群，域擴展。	Roots of polynomials, Galois groups, field extensions.

課程名稱		計算代數	Computational Algebra
課程簡介		格羅布納基，布其柏格算法。	Groebner basis, Buchberger's algorithm.

課程名稱		流形導論	Introduction to Manifolds
課程簡介		流形的局部看起來就像歐氏空間—在這門課裡，我們將給出明確的描述。我們將涵蓋：切向量、光滑性、浸入與淹沒、向量場、向量束、微分形式、以及定向。	A manifold looks like a Euclidean space locally. This will be made precise in the course. We will cover the following topics: tangent vectors, smoothness, immersions and submersions, vector fields, vector bundles, differential forms, and orientations.

課程名稱		數值微分方程導論	Introduction to Numerical Differential Equations
課程簡介		本課程為數值微分方程的入門課程，主要在教導學生如何發展數值方法對微分方程如初始值問題、邊界值問題求解，同時分析數值方法的穩定性與收斂性。	This course provides an introduction to solve ordinary differential equations numerically. The course covers the finite difference approximation, boundary value problems, initial value problems and stability analysis.

課程名稱		傅立葉分析與應用	Fourier Analysis and Applications
課程簡介		許多大自然的現象具有週而復始的規律，即週期現象，由此誘發了偏微分方程的研究。為了求方程的解，Fourier 分析是個重要且不可或缺的工具。這門課是以一個學期為考量，針對理工科系高年級學生而設計的，內容涵蓋Fourier分析與偏微分方程的基本知識與應用。	Many phenomena in nature have the feature of periodic occurence. And thus Fourier analysis is useful in solving problems arising from physics or engineering. In addition to the techniques of solving differential equations, the theoretical aspect is also emphasized in this course so that students can benefit from the study of the idea underlying the subject.

課程名稱		應用機率	Applied Probability
課程簡介		應用於工程、電子、管理、社會科學及數學規劃之基本機率理論與隨機過程。	Elementary Probability Theory and Stochastic Processes applied in field such as engineering, computer science, management science, social science and operations research.

課程名稱		矩陣群	Matrix Groups
課程簡介		本課程為李群暨李代數之預備課程，目標是由例子來解釋李群裡的基本知識。內容涵蓋以下主題: 矩陣群之代數性質，矩陣群上的指數函數暨幾何，以及代數簇。	This is a preliminary course for Lie Algebra vs. Lie groups, aimed to teach students basic knowledge about Lie groups through examples. It covers the following topics: Algebraic properties of matrix groups, the exponential function and the geometry of matrix groups and Algebraic varieties.

課程名稱		賽局論	Game Theory
課程簡介		賽局理論主要研究在博奕行為中的策略決定, 行為中各方是否存在著"最好"方案, 以及如何找到這個方案的數學理論和方法.	Game theory is the study of strategic decision making. Specifically, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." One aim of the course is to teach you some strategic considerations to take into account making your choices. A second aim is to predict how other people or organizations behave when they are in strategic settings. We will see that these aims are closely related. We will learn new concepts, methods and terminology. A third aim is to apply these tools to settings from economics and from elsewhere. The course will provide the basics: representing games and strategies, the extensive form (which computer scientists call game trees), Bayesian games (modeling things like auctions), repeated and stochastic games, and more.

課程名稱		語音訊號處理	Speech Signal Processing
課程簡介		本課程談論聲波、聲波訊號、語音訊號及數位訊號處理。內容包括：(1) 時間域聲波的特徵 (2) 質點振動的常微分方程、特徵值與頻率 (3) 波動方程的邊界值問題 (4) 傅立葉轉換與時頻分析(5)週期函數與傅立葉級數 (6)數位語音合成、識別與診斷	In this course, we will talk about sound waves, acoustic signals, voice signals and digital signal processing. The contents include (1) the features of sound waves in time-domain (2) ordinary differential equations of particle vibration and the corresponding eigenvalues and frequencies (3) boundary value problem of wave equation (4) Fourier transform and time-frequency analysis (5) periodic functions and Fourier series (6) digital speech synthesis, recognition and diagnosis.

課程名稱		數據科學裡的矩陣方法	Matrix Methods in Data Science
課程簡介		在這個課程中，我們將介紹在數據科學與機器學習裡會用到的基本線性代數與矩陣計算方法。課程包括以下幾個部分: 1.介紹資料的基本型態:向量、矩陣、張量，以及相關的基本數學運算。 2.矩陣分解:線性系統，特徵值分解，奇異值分解(SVD)，以及超大型矩陣計算。 3.基本機器學習:主分量分析(PCA)、線性迴歸分析、神經網路。 4.應用：文本數據挖掘、分類、網頁排名等等。	In this course, we will introduce linear algebra and matrix methods in data science and machine learning. The course includes the following parts: 1. Basic Data Types: vectors, matrices, tensors, and related arithmetic. 2. Matrix Factorization: linear systems, eigenvalue decomposition, singular value decomposition (SVD), and large sparse matrix computation. 3. Basic Machine Learning: principal component analysis (PCA), linear regression analysis, neural networks. 4. Applications: classification, text mining, page ranking, etc.