課程名稱 |
應用數學一 Applied Mathematics (Ⅰ) |
開課學期 |
108-2 |
授課對象 |
工學院 應用力學研究所 |
授課教師 |
吳光鐘 |
課號 |
AM7006 |
課程識別碼 |
543EM1020 |
班次 |
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學分 |
3.0 |
全/半年 |
半年 |
必/選修 |
必修 |
上課時間 |
星期二2(9:10~10:00)星期四3,4(10:20~12:10) |
上課地點 |
應111應111 |
備註 |
本課程以英語授課。 總人數上限:98人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1082AM7006_ |
課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
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課程概述 |
一、Syllabus:
I. Introduction to linear spaces
1. linear spaces: linear combination, spanning set, linear dependence,
linear independence, dimension, basis
2. metric spaces: Cauchy sequence, convergent sequence, completeness,
fixed point, contraction mapping, fixed point theorem
3. normed spaces, natural metric, l-p norm, L-p norm
4. inner product spaces: natural norm, Schwartz inequality, Gram-Schmidt
orthgonlization, orthonormal basis, dual bases, adjoint operator, self-
adjoint, eigenvalue problem, eigenexpansion, sets of measure zero
II. Cartesian Tensors
1. Orthonormal Base Vectors
2. Transformation rule of Vectors
3. Scalar, Vector, Pseudo Vector, Pseudo Scalar
4. Dyads, Dyadics, and Tensors
5. Transformation rule of Tensors
6. Quotient Tests
7. Isotropic Tensors
III. Ordinary Differential Equations
1. Initial-Value Problem
2. Existence and Uniqueness Theory
3. System of 1st order ODE’s (const. coefficients)
4. Second-Order ODE
5. Adjoint Operators
6. Green`s Functions and Modified Green`s Function
7. Sturm-Liouville Theory
IV. Partial Differential Equation
1. Introduction
2. Classifications
3. Green`s Function & Integral Representation
4. Other Methods of Solution
5. Maximum-Minimum Principle
二、Prerequisite:
Calculus; Engineering Math (I & II), or Advanced Calculus
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課程目標 |
o This course offers the knowledge to let students
1.understand basic concepts of linear spaces
2.master algebra of Cartesian tensors
3.understand the meaning of existence and uniqueness of linear 1st order system ODE. Master the method of solving linear 1st order system ODE with constant coefficients.
4.master the skill of Green's function and eigen-expansion in solving linear ODE
5.understand the difference among three basic types of linear 2nd order PDE's. Master the skill of Green's function and eigen-expansion in solving linear 2nd order PDEs |
課程要求 |
1.Homeworks
2.Quizzes
3.Mid-term and final exams |
預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
待補 |
參考書目 |
1.Lecture notes
2. H. Jeffreys, "Cartesian tensors," 7th ed., Cambridge Univ. Press, 1968.
3. Y. C. Fung, "A first course in continuum mechanics," Prentice-Hall, 1969.
4. G. Birkho & G. C. Rota, "Ordinary Differential Equations," John Wiley & Sons, 1989.
5. F. Brauer & J. A. Nohel, "Ordinary Differential Equations," Benjamin Inc., 1967.
6. M. W. Hirsch & S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, 1974.
7. I. Stakgold, "Green's Functions and Boundary Value Problems," John Wiley & Sons., 1979.
8. W. E. Williams,“Partial differential equations,” Oxford University Press, 1980.
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評量方式 (僅供參考) |
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週次 |
日期 |
單元主題 |
第1週 |
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Cartesian tensors
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第2週 |
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Cartesian tensors
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第3週 |
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Cartesian tensors
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第4週 |
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Cartesian tensors
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第5週 |
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System of 1st order Ordinary differential equations
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第6週 |
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System of 1st order Ordinary differential equations |
第7週 |
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System of 1st order Ordinary differential equations |
第8週 |
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System of 1st order Ordinary differential equations |
第9週 |
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BVP for 2nd order order Ordinary differential equations |
第10週 |
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BVP for 2nd order order Ordinary differential equations |
第11週 |
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BVP for 2nd order order Ordinary differential equations |
第12週 |
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BVP for 2nd order order Ordinary differential equations |
第13週 |
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Partial differential equations
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第14週 |
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Partial differential equations
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第15週 |
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Partial differential equations
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第16週 |
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Partial differential equations
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第17週 |
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Partial differential equations
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