MIT-工程师数学方法I-课程.doc

工程师数学方法I MIT课程翻译：温秋芳(Wen, Qiufang)(简介并寄信)编辑：何斌(Bin He)(简介并寄信)应用数学导论(Introduction to Applied Mathematics)，Gilbert Strang着， Wellesley-Cambridge出版社。考试考试暂时定在第15, 27，39次课堂上，都是在平时的课上和正常的课程时间。开卷考试，可带书和笔记。没有期末考试。评分作业(9次): 34% 考试 (3次): 66%作业作业中粗体字的练习尤其重要, 典型的表现了本课程中一些额外的深度知识。作业及其完成期限的任何改变都将在课堂上和课程网页上宣布。允许合作完成作业，但是，你必须独立地写出自己的结果，并指出你合作者的名字。不可使用现成的答案。I. 应用线性代数 1. 高斯消去法和轴化2. 矩阵分解 A = LU 和 正定矩阵3. 正定矩阵: 2 2 (省略n n 矩阵的详细证明)4. 最小二乘法: ATA5. 弹簧和质量系统 II. 离散平衡方程 6. 基本平衡方程7. 电网络: AT CA（省略: RLC电路和回路电流）8. 平衡结构: 确定的或不定的9. 不稳定性: 刚性运动及机械装置10. 第19课复习11. 有约束的最小化: 拉各朗日乘数（省略: 投影）12. 对偶性, 能量与共能（co-energy）13. 加权最小二乘法14. 测试1前的复习15. 测试1: 第一、二章III. 连续平衡方程 16. 弹性棒的平衡17. Sturm-Liouville问题，边界层和Delta函数(delta function)18. 弹性梁的平衡（省略: 样条逼近）19. 势流，斯托克斯定理和散度定理20. 格林定理，边界条件和泊松方程21. 变分法: 介绍（省略: 互补最小原理）22. 变分法: 实例（省略:拉各朗日方程和汉密尔顿方程）23. 三维空间的线积分，位势，旋度和梯度 （省略:电磁学）24. 矢量微积分和曲线坐标系25. 流体力学26. 测试2前的复习27. 测试2: 第三章IV. 傅立叶级数和傅立叶变换 28. 傅立叶系数29. 正弦级数和余弦级数, Parseval公式30. 拉普拉斯方程的傅立叶解及其收敛性31. 正交函数: 贝塞尔函数32. 离散傅立叶级数和n单位根33. 卷积定则和信号处理34. 常数对角矩阵35. 傅立叶变换: Plancherel公式和测不准原理36. 变换法则: （省略: 积分方程）37. 常微分方程的解和格林函数38. 测试3前的复习39. 测试3: 第四章 Course Outline Text Strang, Gilbert. Introduction to Applied Mathematics. Wellesley-Cambridge Press. (Table of Contents)ExamsThe exams are tentatively scheduled for Sessions 15, 27 and 39. They are in the normal class and class hour and are OPEN BOOK AND NOTES. There is NO FINAL exam.GradesProblem Sets (9): 34%Exams (3): 66%.Problem SetsExercises in bold in the Problem Sets are especially important and typically present some additional insight on the subject. Any changes to problem sets and due dates will be announced in class and on the course web page. Working together is allowed in problem sets. However, you must write up your results individually and indicate the names of your collaborators. Use of existing solutions is not allowed.I. Applied Linear Algebra 1. Gaussian elimination and pivots2. Factorization A = LU and positive definite matrices3. Positive definite matrices: 2 2 (Omit: detailed proof for n n)4. Least squares: ATA5. Systems of springs and masses II. Equilibrium Equations: Discrete Case 6. Fundamental equations of equilibrium7. Electrical networks: AT CA (Omit: RLC circuit and Loop currents)8. Structures in equilibrium: determinate or indeterminate9. Instability: rigid motion and mechanism10. Review of Lectures 1911. Minimizing with constraints: Lagrange multipliers (Omit: Projections)12. Duality. Energy and co-energy13. Weighted least squares14. Review for Exam 115. EXAM 1: Chapters 1 and 2III. Equilibrium Equations: Continuous Case 16. Equilibrium of an elastic bar17. Sturm-Liouville problem, boundary layers and delta function18. Equilibrium of an elastic beam (Omit: Spline approximations)19. Potential flow, Stokes and divergence theorems20. Greens theorem, boundary conditions and Poissons equation21. Calculus of variations: introduction (Omit: Complementary minimum principle)22. Calculus of variations: examples (Omit: Lagrangians and Hamiltons equation)23. Line integrals, potentials, curl and gradient in 3D (Omit: Electricity and magnetism)24. Vector calculus and curvilinear coordinate systems25. Fluid mechanics26. Review for Exam 227. EXAM 2: Chapter 3IV. Fourier Series and Transforms 28. Fourier coeffiients29. Sine and cosine series, Parsevals formula30. Fourier solution to Laplace equation and convergence31. Orthogonal functions; Bessel functions32. Discrete Fourier series and the n roots of unity33. Convolution rule and signal processing34. Constant-diagonal matrices35. Fourier transforms: Plancherels formula and uncertainty principle36. Transform rules (Omit: Integral equations)37. Solutions of ODEs and Greens function38. Review for Exam 339. EXAM 3: Chapter 4 目录INTRODUCTION TO APPLIED MATHEMATICSGilbert StrangWellesley-Cambridge Press (1986)TABLE OF CONTENTS 1. Symmetric Linear Systems1.1 Introduction1.2 Gaussian Elimination1.3 Positive Definite Matrices1.4 Minimum Principles1.5 Eigenvalues and Dynamical Systems1.6 A Review of Matrix Theory 2. Equilibrium Equations2.1 A Framework for the Applications2.2 Constraints and Lagrange Multipliers2.3 Electrical Networks2.4 Structures in Equilibrium2.5 Least Squares Estimation and the Kalman Filter 3. Equilibrium in the Continuous Case3.1 One-dimensional Problems3.2 Differential Equations of Equilibrium3.3 Laplaces Equation and Potential Flow3.4 Vector Calculus in Three Dimensions3.5 Equilibrium of Fluids and Solids3.6 Calculus of Variations 4. Analytical Methods4.1 Fourier Series and Orthogonal Expansions4.2 Discrete Fourier Series and Convolution4.3 Fourier Integrals4.4 Complex Variables and Conformal Mapping4.5 Complex Integration 5. Numerical Methods5.1 Linear and Nonlinear Equations5.2 Orthogonalization and Eigenvalue Problems5.3 Semi-direct and Iterative Methods5.4 The Finite Element Method5.5 The Fast Fourier Transform 6. Initial-Value Problems6.1 Ordinary Differential Equations6.2 Stability and the Phase Plane and Chaos6.3 The Laplace Transform and the z-Transform6.4 The Heat Equation vs. the Wave Equation6.5 Difference Methods for Initial-Value Problems6.6 Nonlinear Conservation Laws 7. Network Flows and Combinatorics7.1 Spanning Trees and Shortest Paths7.2 The Marriage Problem7.3 Matching Algorithms7.4 Maximal Flow in a Network 8. Optimization8.1 In