課程資訊
課程名稱
工程數學下
Engineering Mathematics (2) 
開課學期
102-2 
授課對象
機械工程學系  
授課教師
伍次寅 
課號
ME2002 
課程識別碼
502 20002 
班次
02 
學分
全/半年
全年 
必/選修
必修 
上課時間
星期一3,4(10:20~12:10)星期三2(9:10~10:00) 
上課地點
工綜215工綜B01 
備註
限本系所學生(含輔系、雙修生)
總人數上限:65人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1022ME2002_02 
課程簡介影片
 
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課程大綱
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課程概述

數學是所有理工學科的基礎。大凡對於任何一門科學領域中的問題,從一開始的現象描述、建構模式到最後分析求解、得出結論的過程,大概都脫不了要使用到數學。工程數學是所有工科學生所必須修習的一門基礎知識課程,它除了延續同學們在大一所學過的微積分觀念與知識之外,並包括了分析一般工程問題所需應用到的基本數學工具,以及求解數學模式所需使用到的方法與技巧。這門課除了介紹數學方面的知識,亦強調其方法與應用。為了不讓學生覺得學了半天數學卻不知為何而學以及用於何處,本課程將盡量避開不必要的理論證明,並從各位同學接下來即將要修習的各個知識領域的課程諸如材料力學、流體力學、熱傳學、自動控制、電路學、系統動力、振動學、訊號處理…..當中舉例來說明工程數學的應用。工程數學(下)的內容主要為向量函數的微積分、傅立業(Fourier)級數展開及傅立業轉換、偏微分方程式及其解,以及複變數分析與應用。課程會使用到相當多函數的微分與積分(特別是積分),同學們事先應將大一的微積分溫習一遍。另外,除了偏微分方程式的部分會應用到工程數學(上)中之常微分方程式的解與技巧,其他的章節基本上與工程數學(上)無關,同學可不必太過於擔心課程銜接上的問題。
本課程的後續課程計有:工程數學二、高等工程數學、偏微分方程式、工程統計法、數值分析、擾動學…..等。同學們若有興趣的話亦可至外系或數學系選修更專門的數學課程,以增進數學方面知識的廣度與深度。

本課程授課大綱如下:

1. Vector Calculus:
vector functions, differentiation of vector functions, curves and surfaces, tangents and normals, integration of vector functions, line integrals, surface integrals, gradient, divergence and curl, Green`s theorem, Gauss` theorem, Stokes` theorem, potential theory

2. Fourier Analysis:
Fourier series, convergence of Fourier series, Fourier integrals, Fourier transforms, properties of Fourier transform, complex Fourier transforms, Fourier spectra, power spectra, discrete Fourier transform

3. Orthogonal Expansions: (if time permits)
special functions (Legendre, Bessel), orthogonal polynomials, Sturm-Liouville theory, eigenfunction expansions

4. Linear Partial Differential Equations:
method of separation of variables, eigenvalue problems, eigen-solutions, heat equations, wave equations, Laplace equations, non-homogeneous equations and boundary conditions, Laplace and Fourier transforms for solving boundary-value problems

5. Complex Analysis:
complex variables and complex functions, analytic functions, differentiation and integration of complex functions, multi-valued functions, branch point and branch cut, Cauchy integral theorem, power series, Taylor and Laurent series representation of functions, singularities, residue theorem, application on real integrals
 

課程目標
傳授學生修習理工學科所必需具備的數學知識與解析工具,並訓練學生能將之應用於求解一般工程問題之數學模式上。 
課程要求
待補 
預期每週課後學習時數
 
Office Hours
備註: Wed. 12:00~2:00 pm office #618, Engineering building 
參考書目
'Advanced Engineering Mathematics', 7th ed. (custom publishing),
P. V. O’Neil, Cengage, 2012.
 
指定閱讀
待補 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
6次小考 
70% 
 
2. 
期末考 
40% 
 
3. 
作業 
0% 
指定各章節習題,但無須繳交,亦不予評分。 
4. 
(1)+(2)=學期成績 
110% 
原始分數將轉換成等第制  
 
課程進度
週次
日期
單元主題
第1週
2/17,2/19  Vector differential calculus --
vector functions, curves (length, tangent, normal, curvature and torsion), Frenet formulas, velocity & acceleration of particle 
第2週
2/24,2/26  Vector differential calculus --
vector field, streamlines, gradient of scalar field, directional derivative, physical interpretation of gradient, divergence & curl of vector field, physical interpretation of divergence & curl 
第3週
3/03,3/05  Vector differential calculus --
important vector identities;
Vector integral calculus --
line integrals of scalar & vector fields, conservative vector field & potential theory, representations of surfaces, surface area 
第4週
3/10,3/12  Vector integral calculus --
surface integral of scalar field, surface integral of vector field, flux across a surface;
** 1st quiz (tentative schedule) 
第5週
3/17,3/19  Vector integral calculus --
Green’s theorem, extension of Green’s theorem, divergence (Gauss) theorem, Green’s identities, Stokes theorem, Stokes theorem & potential theory 
第6週
3/24,3/26  Fourier series & Fourier Integrals --
Fourier series (concept & idea), Fourier coefficients, Fourier-series representation of functions, piecewise continuous & piecewise smooth functions 
第7週
3/31,4/02  Fourier series & Fourier Integrals --
convergence properties of Fourier series;
** 2nd quiz (tentative schedule);
* no class on 4/2 (spring break)
 
第8週
4/07,4/09  Fourier series & Fourier Integrals --
Fourier cosine & sine series, differentiation & integration of Fourier series, Parseval theorem, Fourier-series representation of periodic functions, Fourier spectrum, complex Fourier series;
Fourier series & Fourier Integrals --
Fourier integrals, convergence of Fourier integral 
第9週
4/14,4/16  Fourier series & Fourier Integrals --
Fourier cosine & sine integrals, complex Fourier integrals;
Fourier transforms --
idea of Fourier transform, Fourier transform & inverse Fourier transform, Fourier spectrum;
** 3rd quiz (tentative schedule) 
第10週
4/21,4/23  Fourier transforms --
properties of Fourier transform, convolution, applications of Fourier transform, windowed Fourier transform, filtering & Dirac delta function, lowpass & bandpass filters, Fourier cosine & sine transforms, discrete Fourier transform, power spectrum;
Partial differential equations (PDE) --
types of PDE (heat equation, wave equation, Laplace equation) 
第11週
4/28,4/30  Partial differential equations (PDE) --
methods of separation of variables, solutions to unsteady heat equation with different boundary conditions;
** 4th quiz (tentative schedule) 
第12週
5/05,5/07  Partial differential equations (PDE) --
unsteady heat conduction in infinite media, steady heat equation (Laplace equation), Laplace equation in polar coordinates, Laplace equation for unbounded domain 
第13週
5/12,5/14  Partial differential equations (PDE) --
wave equation, wave equation for infinite string, d’Alembert solution of wave equation;
Partial differential equations (PDE) --
heat equation in cylindrical and spherical coordinates 
第14週
5/19,5/21  Partial differential equations (PDE) --
equations in multiple spatial coordinates, non-homogeneous equation & non-homogeneous boundary conditions;
** 5th quiz (tentative schedule)
 
第15週
5/26,5/28  Partial differential equations (PDE) --
method of Laplace transform, method of Fourier transforms;
Complex analysis --
complex variables & complex functions, Cauchy-Riemann conditions, analytic functions, power series, complex exponential & trigonometric functions, multi-valued functions (complex logarithm & fractional powers), branch point & branch cut 
第16週
6/02,6/04  * no class on 6/2 (national holiday);
** 6th quiz (tentative schedule) 
第17週
6/09,6/11  Complex integrations --
Cauchy integral theorem, Cauchy integral formula, power series representation of function (Taylor & Laurent series), singularities (poles), residue theorem, application of residue theorem, evaluation of real integrals using complex integration techniques 
第18週
6/16  ** 6/16 final exam (according to school’s schedule)