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課程名稱 |
應用數學上
Applied Mathematics (1) |
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開課學期 |
114-2 |
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授課對象 |
理學院 物理學系 |
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授課教師 |
吳俊輝 |
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課號 |
Phys2024 |
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課程識別碼 |
202 20411 |
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班次 |
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學分 |
3.0 |
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全/半年 |
全年 |
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必/選修 |
必帶 |
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上課時間 |
星期二8,9,10(15:30~18:20) |
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上課地點 |
新物111 |
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備註 |
限本系所學生(含輔系、雙修生)
總人數上限：80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 |
(本課程不加簽外系學生, 請勿一直寫信或找授課教師要求加簽)
This course is the first part of two which provide the foundation of mathematical methods commonly needed in the study of physics. |
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課程目標 |
This course is designed to help physics majors become familiar with fundamental mathematical tools used across a broad range of topics in physics. This is the first semester of a two-semester sequence. Its covers the following topics: Linear Algebra, Fourier series and Fourier transform, ODE, Vector calculus, PDE, Green’s functions. |
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課程要求 |
There will be a midterm and a final exam, scheduled according to the university calendar.
Homework will not be graded but will be supported by the TA.
Attendance is not mandatory. |
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預期每週課前或/與課後學習時數 |
Students are expected to spend approximately 2 hours preparing before each week’s lectures and 3 hours reviewing afterward. |
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Office Hours |
每週五 13:20~14:10 備註： (本課程不加簽外系學生, 請勿一直寫信或找授課教師要求加簽)
新物 111
助教：
晏千博 f13222037@ntu.edu.tw
劉雨恩 r13222079@ntu.edu.tw |
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指定閱讀 |
The lecture materials are compiled from various books in the reference list. The students should refer to whatever suitable for their needs. |
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參考書目 |
G. Arfken and H. J. Weber (2005). Mathematical Methods for Physi- cists, 6th edition. Academic Press.
Strang/ Introduction to Linear Algebra International Edition.
E Kreyszig (2011). Advanced Engineering Mathematics, 8th edition. Wiley (10th edition available).
K F Riley, M P Hobson & S J Bence (2002). Mathematical Methods for Physics and Engineering. 3rd ed., Cambridge University Press. (Available online via http://idiscover.lib.cam.ac.uk).
J W Dettman (1988). Mathematical Methods in Physics and Engineering. Dover (Dover Books on Physics). |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Mid-term exam |
50% |
The exam date is scheduled according to the university calendar, unless otherwise announced. The exam time is the normal lecture time.
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2. |
Final exam |
50% |
The exam date is 6/2 (Tuesday), the normal lecture time. |
- 本校尚無訂定 A+ 比例上限。
- 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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週次 |
日期 |
單元主題 |
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第1-16週 |
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(本課程不加簽外系學生, 請勿一直寫信或找授課教師要求加簽)
Linear Algebra (6 weeks)
Linear vector spaces. Matrices, determinants and traces. Minors, cofactors, and inverse of a matrix. Orthogonal transformation, rotations, reflections, similarity transformation, and Wronskian. Hermitian, anti-Hermitian, unitary, adjoint operator, and unitary transformation. Inner product (scalar product) and other products of vectors and matrices. Linear equations. Diagonalisation, eigenvalues and eigenvectors. Quadratic and Hermitian forms. Stationary property of the eigenvalues. Orthogonality of functions. Integral transform. Solutions for system of ODEs. Transfer functions of linear nth-order ODE.
ODE (3 weeks)
Degrees/orders, IC/BC, direction field, First order ODE (separable equations; change of variables; linear equations; exact equations; integrating factors; non-linear/non-uniqueness; iteration method; examples involving substitution), Second-order linear ODE (constant coefficients; homogeneous/nonhomogeneous; Cauchy-Euler (C-E) equations; exp(ax) as trial solution, including degenerate case), Superposition, Particular integrals and complementary functions, Constants of integration and number of necessary boundary/initial conditions, Particular integrals by trial solutions, Examples including radioactive sequences, Resonance, transients and damping. Higher-order ODE (constant coefficients; variation of parameters).
Vector calculus (1 week)
Suffix notation. Einstein summation convention. Contractions using δij and εijk. Reminder of vector products, grad, div, curl, del2, and their representations using suffix notation. Vector differential operators in orthogonal curvilinear coordinates (e.g. cylindrical and spherical polar coordinates. Jacobians). Integral theorems (gradient theorem, divergent/Gauss’s theorem, curl/Stokes’s theorem).
PDE (1 week)
Linear second-order partial differential equations; physical examples of occurrence (Laplace’s/Poisson’s equations, diffusion equation, wave equation, Helmholtz equation), verification of solution by substitution. Linear superposition. Examples of nonlinear PDE’s. Method of separation of variables (Cartesian coordinates only).
Green’s functions (1 week)
Response to impulses, delta function (treated heuristically), Green's functions for initial and boundary value problems. Treatment of inhomogeneous linear 2nd-order ODE.
Laplace and Poisson's equations (2 weeks)
Solution by separation of variables of Laplace's equation in plane polar coordinates, and spherical polar coordinates (axisymmetric case); Legendre polynomials again. Solution of Poisson's equation as an integral. Uniqueness for Poisson's equation with Dirichlet boundary conditions. Green's identity. Green's function for Laplace's equation with simple boundary conditions using the method of images. Applications to electrostatic fields and steady heat flow.
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