数学课程是基础，Coursera上有很多数学公开课，这里做个汇总，注意由于Coursera上有一批很有特色的统计学相关的数学课程，我们将在下一期里单独汇总。
1 斯坦福大学 Introduction to Mathematical Thinking（数学思维导论）
http://coursegraph.com/coursera-mathematical-thinking
引用老版课程一个同学的评价，供参考：
这门课是高中数学到大学数学的一个过度。高中数学一般重计算不太注重证明，这门课讲了基本的逻辑，数学语言（两个 quantifier，there exists， for all）和证明的几个基本方法，比如证明充要条件要从两个方向证、证伪只需要举个反例，原命题不好证的时候可以证等价的逆否命题以及很常用的数学归纳法。课程讲了数论里一些基本定理，然后通过让你证一些看起来显然而不需要证明的证明题来训练你证明的技能和逻辑思考的能力，看起来显然的命题也是要证明才能说服人的，课程最后简略的讲了下数学分析里面实数的引入，但这部分讲的不完整。Keith Devlin 是个 old school 的讲师，上课只用纸和笔，也是属于比较热情的讲师，他每周都会录几个答疑的视频。这门比较适合大一的新生上，开得也比较频繁。
课程简介：
Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years. Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
2 加州大学尔湾分校 初级微积分系列课程
1）Pre-Calculus: Functions（初级微积分：函数）
http://coursegraph.com/coursera-pre-calculus
This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Solve linear and quadratic equations 2. Solve some classes of rational and radical equations 3. Graph polynomial, rational, piece-wise, exponential and logarithmic functions 4. Find integer roots of polynomial equations 5. Solve exponential and logarithm equations 6. Understand the inverse relations between exponential and logarithm equations 7. Compute values of exponential and logarithm expressions using basic properties
2）Pre-Calculus: Trigonometry(初级微积分：三角）
http://coursegraph.com/coursera-trigonometry
This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of triangles. This course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Evaluate trigonometric functions using the unit circle and right triangle approaches 2. Solve trigonometric equations 3. Verify trigonometric identities 4. Prove and use basic trigonometric identities. 5. Manipulate trigonometric expressions using standard identities 6. Solve right triangles 7. Apply the Law of Sines and the Law of Cosines
3 宾夕法尼亚大学的 单变量微积分系列课程
1）Calculus: Single Variable Part 1 – Functions（单变量微积分1：函数）
http://coursegraph.com/coursera-single-variable-calculus
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this first part–part one of five–you will extend your understanding of Taylor series, review limits, learn the *why* behind l’Hopital’s rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O.
2）Calculus: Single Variable Part 2 – Differentiation（单变量微积分2：微分）
http://coursegraph.com/coursera-differentiation-calculus
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this second part–part two of five–we cover derivatives, differentiation rules, linearization, higher derivatives, optimization, differentials, and differentiation operators.
3）Calculus: Single Variable Part 3 – Integration（单变量微积分3：积分）
http://coursegraph.com/coursera-integration-calculus
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this third part–part three of five–we cover integrating differential equations, techniques of integration, the fundamental theorem of integral calculus, and difficult integrals.
4) Calculus: Single Variable Part 4 – Applications（单变量微积分4：应用）
http://coursegraph.com/coursera-applications-calculus
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this fourth part–part four of five–we cover computing areas and volumes, other geometric applications, physical applications, and averages and mass. We also introduce probability.
4 杜克大学 Data Science Math Skills（数据科学中的数学技巧）
http://coursegraph.com/coursera-datasciencemathskills
这门课程主要介绍数据科学中涉及的相关数学概念，让学生了解基本的数学概念，掌握基本的数学语言，内容涵盖集合论、求和的Sigma符号、数学上的笛卡尔（x，y）平面、指数、对数和自然对数函数，概率论以及叶斯定理等：
Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists must know before moving on to more advanced material.
5 加州大学圣迭戈分校 Introduction to Discrete Mathematics for Computer Science Specialization（面向计算机科学的离散数学专项课程)
http://coursegraph.com/coursera-specializations-discrete-mathematics
面向计算机科学的离散数学专项课程（Introduction to Discrete Mathematics for Computer Science Specialization），这个系列包含5门子课程，涵盖证明、组合数学与概率、图论，数论和密码学，配送问题项目等，感兴趣的同学可以关注: Build a Foundation for Your Career in IT-Master the math powering our lives and prepare for your software engineer or security analyst career
Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and financial analysts (it is not a coincidence that math puzzles are often used for interviews). We cover the basic notions and results (combinatorics, graphs, probability, number theory) that are universally needed. To deliver techniques and ideas in discrete mathematics to the learner we extensively use interactive puzzles specially created for this specialization. To bring the learners experience closer to IT-applications we incorporate programming examples, problems and projects in our courses.
http://coursegraph.com/coursera-what-is-a-proof
There is a perceived barrier to mathematics: proofs. In this course we will try to convince you that this barrier is more frightening than prohibitive: most proofs are easy to understand if explained correctly, and often they are even fun. We provide an accompanied excursion in the “proof zoo” showing you examples of techniques of different kind applied to different topics. We use some puzzles as examples, not because they are “practical”, but because discussing them we learn important reasoning and problem solving techniques that are useful. We hope you enjoy playing with the puzzles and inventing/understandings the proofs. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.
2）Combinatorics and Probability（组合和概率）
Counting is one of the basic mathematically related tasks we encounter on a day to day basis. The main question here is the following. If we need to count something, can we do anything better than just counting all objects one by one? Do we need to create a list of all phone numbers to ensure that there are enough phone numbers for everyone? Is there a way to tell that our algorithm will run in a reasonable time before implementing and actually running it? All these questions are addressed by a mathematical field called Combinatorics. In this course we discuss most standard combinatorial settings that can help to answer questions of this type. We will especially concentrate on developing the ability to distinguish these settings in real life and algorithmic problems. This will help the learner to actually implement new knowledge. Apart from that we will discuss recursive technique for counting that is important for algorithmic implementations. One of the main `consumers’ of Combinatorics is Probability Theory. This area is connected with numerous sides of life, on one hand being an important concept in everyday life and on the other hand being an indispensable tool in such modern and important fields as Statistics and Machine Learning. In this course we will concentrate on providing the working knowledge of basics of probability and a good intuition in this area. The practice shows that such an intuition is not easy to develop. In the end of the course we will create a program that successfully plays a tricky and very counterintuitive dice game. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.
3）Introduction to Graph Theory（图论导论）
We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. In this course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible! By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.
4) Number Theory and Cryptography（数论和密码学）
We all learn numbers from the childhood. Some of us like to count, others hate it, but any person uses numbers everyday to buy things, pay for services, estimated time and necessary resources. People have been wondering about numbers’ properties for thousands of years. And for thousands of years it was more or less just a game that was only interesting for pure mathematicians. Famous 20th century mathematician G.H. Hardy once said “The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics”. Just 30 years after his death, an algorithm for encryption of secret messages was developed using achievements of number theory. It was called RSA after the names of its authors, and its implementation is probably the most frequently used computer program in the word nowadays. Without it, nobody would be able to make secure payments over the internet, or even log in securely to e-mail and other personal services. In this short course, we will make the whole journey from the foundation to RSA in 4 weeks. By the end, you will be able to apply the basics of the number theory to encrypt and decrypt messages, and to break the code if one applies RSA carelessly. You will even pass a cryptographic quest! As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.
5）Solving Delivery Problem（解决旅行商问题）
http://coursegraph.com/coursera-delivery-problem
We’ll implement together an efficient program for a problem needed by delivery companies all over the world millions times per day — the travelling salesman problem. The goal in this problem is to visit all the given places as quickly as possible. How to find an optimal solution to this problem quickly? We still don’t have provably efficient algorithms for this difficult computational problem and this is the essence of the P versus NP problem, the most important open question in Computer Science. Still, we’ll implement several efficient solutions for real world instances of the travelling salesman problem. While designing these solutions, we will rely heavily on the material learned in the courses of the specialization: proof techniques, combinatorics, probability, graph theory. We’ll see several examples of using discrete mathematics ideas to get more and more efficient solutions.
6 伦敦帝国理工学院 Mathematics for Machine Learning Specialization（面向机器学习的数学专项课程系列）
http://coursegraph.com/coursera-specializations-mathematics-machine-learning
伦敦帝国理工学院的面向机器学习的数学专项课程系列（Mathematics for Machine Learning Specialization），该系列包含3门子课程，涵盖线性代数，多变量微积分，以及主成分分析（PCA），这个专项系列课程的目标是弥补数学与机器学习以及数据科学鸿沟，感兴趣的同学可以关注：Mathematics for Machine Learning。Learn about the prerequisite mathematics for applications in data science and machine learning
For a lot of higher level courses in Machine Learning and Data Science, you find you need to freshen up on the basics in maths – stuff you may have studied before in school or university, but which was taught in another context, or not very intuitively, such that you struggle to relate it to how it’s used in Computer Science. This specialisation aims to bridge that gap, getting you up to speed in the underlying maths, building an intuitive understanding, and relating it to Machine Learning and Data Science. In the first course on Linear Algebra we look at what linear algebra is and how it relates to data. Then we look through what vectors and matrices are and how to work with them. The second course, Multivariate Calculus, builds on this to look at how to optimise fitting functions to get good fits to data. It starts from introductory calculus and then uses the matrices and vectors from the first course to look at data fitting. The third course, Dimensionality Reduction with Principal Components Analysis, uses the maths from the first two courses to do simple optimisation for the situation where you don’t have an understanding of how the data variables relate to each other. At the end of this specialisation you will have gained the prerequisite mathematical knowledge to continue your journey and take more advanced courses in machine learning.
1） Mathematics for Machine Learning: Linear Algebra（面向机器学习的数学：线性代数）
http://coursegraph.com/coursera-linear-algebra-machine-learning
In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets – like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we’re aiming at data-driven applications, we’ll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you’ll write code blocks and encounter Jupyter notebooks in Python, but don’t worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning.
2）Mathematics for Machine Learning: Multivariate Calculus（面向机器学习的数学：多变量微积分）
http://coursegraph.com/coursera-multivariate-calculus-machine-learning
This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. We start at the very beginning with a refresher on the “rise over run” formulation of a slope, before converting this to the formal definition of the gradient of a function. We then start to build up a set of tools for making calculus easier and faster. Next, we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We take a look at how we can use calculus to build approximations to functions, as well as helping us to quantify how accurate we should expect those approximations to be. We also spend some time talking about where calculus comes up in the training of neural networks, before finally showing you how it is applied in linear regression models. This course is intended to offer an intuitive understanding of calculus, as well as the language necessary to look concepts up yourselves when you get stuck. Hopefully, without going into too much detail, you’ll still come away with the confidence to dive into some more focused machine learning courses in future.
3）Mathematics for Machine Learning: PCA（面向机器学习的数学：主成分分析）
http://coursegraph.com/coursera-pca-machine-learning
This course introduces the mathematical foundations to derive Principal Component Analysis (PCA), a fundamental dimensionality reduction technique. We’ll cover some basic statistics of data sets, such as mean values and variances, we’ll compute distances and angles between vectors using inner products and derive orthogonal projections of data onto lower-dimensional subspaces. Using all these tools, we’ll then derive PCA as a method that minimizes the average squared reconstruction error between data points and their reconstruction. At the end of this course, you’ll be familiar with important mathematical concepts and you can implement PCA all by yourself. If you’re struggling, you’ll find a set of jupyter notebooks that will allow you to explore properties of the techniques and walk you through what you need to do to get on track. If you are already an expert, this course may refresh some of your knowledge.
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