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課程名稱 |
工程數學上
Engineering Mathematics (1) |
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開課學期 |
115-1 |
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授課對象 |
機械工程學系 |
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授課教師 |
何亦平 |
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課號 |
ME2001 |
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課程識別碼 |
502E20001 |
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班次 |
02 |
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學分 |
3.0 |
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全/半年 |
全年 |
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必/選修 |
必修 |
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上課時間 |
星期一3,4(10:20~12:10)星期三2(9:10~10:00) |
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上課地點 |
機械114機械114 |
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備註 |
本課程以英語授課。
限本系所學生(含輔系、雙修生)
總人數上限：55人 |
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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 |
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課程大綱
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課程概述 |
Engineering Mathematics I is the first semester of a two-semester engineering mathematics sequence for mechanical engineering undergraduates. It develops analytical tools for modeling and solving physical systems, covering first- and higher-order ordinary differential equations (ODEs), Laplace transforms, series solutions of linear ODEs, linear algebra, and systems of linear first-order ODEs. Applications including thermal relaxation, vibration, viscoelasticity, circuits, and related engineering problems are used throughout. Semester 2 covers Fourier series and transforms, partial differential equations, and complex analysis.
***In-person lectures unless otherwise noted
***Lecture recording will only be provided for pre-approved cases
***Lectures in English
***Midterms and Final Exam are to be held in the regular lecture hours |
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課程目標 |
By the end of the semester, students are expected to:
- Solve first- and higher-order ODEs analytically using standard techniques;
- Apply Laplace transforms to solve linear ODEs and coupled systems;
- Obtain power-series and Frobenius solutions about ordinary and singular points;
- Perform fluent matrix computations including row reduction, determinants, and eigenvalue/eigenvector analysis;
- Solve systems of linear first-order ODEs by the eigenvalue method and interpret the phase portraits of homogeneous linear systems;
- Model engineering problems (including thermal relaxation, vibration, RLC circuits, viscoelastic response, and reaction/growth kinetics) using ODEs; state and justify assumptions; and verify important results independently;
- Use generative AI critically: frame and solve the core mathematics independently, ask AI to solve the same problem, audit its reasoning, and reach a defensible conclusion through independent verification; this process is practiced in selected class activities and assessed in the Final Project. |
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課程要求 |
Please review your calculus, especially the sections on transcendental and/or hyperbolic functions, e.g., sinh(x), cosh(x), tanh(x), etc. Familiarity with these functions will be helpful.
Suggested Reading: Stewart, Clegg, and Watson, Calculus Early Transcendentals, 9th edition. Chap 1.4, 1.5, 3.11 (and Chap 9 on differential equations). |
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預期每週課前或/與課後學習時數 |
3-5 hours at least |
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Office Hours |
另約時間 |
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指定閱讀 |
o P. V. O'Neil, Advanced Engineering Mathematics, 8th Edition (SI Edition). Cengage Learning, 2018. ISBN 9789579282031. [Primary textbook] |
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參考書目 |
o D. G. Zill, Advanced Engineering Mathematics, 7th Edition. Jones & Bartlett Learning, 2017. [Reference textbook, complementary reading and additional practice problems]
o O’Neil’s online web modules — Spring Motion, Review of Partial Fractions Decomposition, Equations with Polynomial Coefficients, and Least Squares Vectors and Data Fitting — accompany the textbook and are used in the sessions of Oct. 7 and Oct. 19 and may also be useful for the Final Project. Access instructions will be posted on NTU COOL. |
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評量方式
(僅供參考) |
- 本校建議 A+ 比例上限為 20% ，非強制規定，授課教師可依課程要求調整，建議必修課程參考。
- 本校採用等第制評定成績，學生成績評量辦法中的百分制分數區間與單科成績對照表僅供參考，授課教師可依等第定義調整分數區間。詳見學習評量專區 (連結)。
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針對學生困難提供學生調整方式 |
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上課形式 |
以錄音輔助 |
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作業繳交方式 |
學生與授課老師協議改以其他形式呈現 |
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考試形式 |
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其他 |
由師生雙方議定 |
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週次 |
日期 |
單元主題 |
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第1週 |
Sep. 7 (M) |
Course introduction; syllabus and AI-use policy walk-through. Terminology, classification, initial-value problems; separable equations; Singular solutions |
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第1週 |
Sep. 9 (W) |
The linear first-order equation; integrating-factor solution. MSE examples: Newton’s law of cooling, RC charging |
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第2週 |
Sep. 14 (M) |
Exact equations; test for exactness; recovering the potential function. Integrating factors for non-exact equations; Homogeneous, Bernoulli and Riccati equations — substitution methods |
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第2週 |
Sep. 16 (W) |
The linear second-order equation: superposition, linear independence, the Wronskian, structure of the general solution |
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第3週 |
Sep. 21 (M) |
Reduction of order; The constant-coefficient homogeneous equation; nth-order generalization |
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第3週 |
Sep. 23 (W) |
Particular solutions of the nonhomogeneous equation: the method of undetermined coefficients |
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第4週 |
Sep. 28 (M) |
HOLIDAY (No Class) |
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第4週 |
Sep. 30 (W) |
Undetermined coefficients |
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第5週 |
Oct. 5 (M) |
Variation of parameters; Euler’s differential equation |
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第5週 |
Oct. 7 (W) |
Applications: spring–mass–damper systems and damping regimes; resonance; RLC circuits; viscoelastic models (Maxwell, Kelvin–Voigt); two-point boundary-value problems. |
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第6週 |
Oct. 12 (M) |
Solution Audit/Midterm 1 Preparation
Part 3 begins — Laplace transform: definition and notation, existence conditions, linearity, transforms of elementary functions. |
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第6週 |
Oct. 14 (W) |
MIDTERM 1 — Parts 1 and 2 (First-Order ODEs + Higher-Order Linear ODEs). 50-min closed-book exam. Includes one Solution Audit item. |
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第7週 |
Oct. 19 (M) |
The inverse Laplace transform; partial fractions; completing the square; Solution of initial-value problems; transforms of derivatives; the t-space ↔ s-space picture; equations with polynomial coefficients. |
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第7週 |
Oct. 21 (W) |
The first shifting theorem; the Heaviside function H(t − a), pulses, and the second shifting theorem. |
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第8週 |
Oct. 26 (M) |
HOLIDAY (No Class) |
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第8週 |
Oct. 28 (W) |
Heaviside’s formula; piecewise-defined forcing — worked examples (switched RLC, step-loaded viscoelastic element); periodic forcing and the staircase function. |
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第9週 |
Nov. 2 (M) |
Convolution; the transfer function and impulse response; Impulses and the Dirac delta function; systems of linear differential equations by Laplace transform (revisited in Part 5). |
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第9週 |
Nov. 4 (W) |
Power series review; radius of convergence; analytic functions; ordinary versus singular points; the power series method and recurrence relations. |
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第10週 |
Nov. 11 (W) |
Solution Audit/Midterm 2 Preparation
Vectors in the plane and 3-space;
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第10週 |
Nov. 9 (M) |
Power series solutions — worked examples (Airy equation); MSE context; Frobenius solutions — regular singular points, the indicial equation, and the three cases of indicial roots. |
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第11週 |
Nov. 16 (M) |
Lines and planes; the dot product and projections; the cross product; n-vectors and the algebraic structure of Rⁿ; subspaces, spanning sets, linear independence, basis and dimension. |
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第11週 |
Nov. 18 (W) |
MIDTERM 2 — Parts 3 and 4 (Laplace Transform + Series Solutions). 50-min closed-book exam; Laplace table provided |
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第12週 |
Nov. 23 (M) |
Orthogonal sets and the Gram–Schmidt process; orthogonal complements and projections; Matrices and matrix algebra; special matrices; matrix multiplication revisited; application to random walks in crystals |
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第12週 |
Nov. 25 (W) |
Row operations and reduced matrices; Gaussian and Gauss–Jordan elimination; rank and nullity |
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第13週 |
Dec. 2 (W) |
Prerecorded Lecture/online Q&A: Matrix inverses; determinants and evaluation by row and column operations; Cramer’s rule |
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第13週 |
Nov. 30 (M) |
Prerecorded Lecture/online Q&A: Solution of homogeneous linear systems Ax = 0; null space and dimension of the solution space; Solution of nonhomogeneous linear systems Ax = b; consistency, uniqueness, parameterized solutions |
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第14週 |
Dec. 7 (M) |
Eigenvalues and eigenvectors; the characteristic equation; eigenspaces; linear independence of eigenvectors; Diagonalization; special matrices |
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第14週 |
Dec. 9 (W) |
Systems of linear first-order ODEs; structure of solutions of X′ = AX; solution for constant A by the eigenvalue method, including the complex eigenvalue case; phase portraits of homogeneous linear systems. |
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第15週 |
Dec. 14 (M) |
PROJECT PRESENTATIONS (both sessions; running order to be announced) |
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第15週 |
Dec. 16 (W) |
Final-exam review; Q&A; closing remarks. |
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第16週 |
Dec. 21 (M) |
FINAL EXAM (10:20 – 12:00, 100 min, closed-book), Parts 1 to 5. Laplace table provided. |
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