Goals of the Course(JPN)
Goals of the Course
|This course has dual aims: 1) to convey mathematical principles relevant to solving applied problems in physics, engineering and chemistry; 2) to improve students’ technical ability – i.e. ability to express intuition in mathematical terms and ability to solve problems.
At the end of this course, students will have mastered a variety of first order linear and nonlinear ODEs, second and arbitrary order linear ODEs and their solution methods, Fourier series, the Fourier transform, convolution and the Laplace transform together with applications.
Students will find this mathematical methods course helpful in other units such as Quantum Mechanics, Analytical Mechanics, Electricity and Magnetism, as well as in Chemistry, Automotive Engineering and other engineering courses.
This course is a companion course to Mathematical Physics II.
Objectives of the Course(JPN)
Objectives of the Course
|Students master analytical techniques for problems that arise in physics, engineering and chemistry. This course introduces first order and second order ordinary differential equations and their solution methods. Questions of uniqueness of solutions and convergence are also discussed. Students are also introduced to Fourier series, the Fourier transform, convolution, Laplace transform, and the Dirac delta function, and applications of these.|
Course Content / Plan
|• First order ordinary differential equation (ODE) initial value problems. Integration factor; separable equations; systems of ODEs (Hamiltonian systems); phase plane, flow. Uniqueness and existence theorems. Some differences between linear and nonlinear ODEs.
• Second order linear ODE initial value problems. Homogeneous solution. Proving linear independence (Wronskian). Method of Undetermined Coefficients; Variation of Parameters. Series solutions: ordinary point, regular singular point; convergence tests; Method of Frobenius. Examples from physics, engineering and chemistry.
• Fourier series. Dirichlet conditions. Role of symmetry. Gibbs phenomenon. Effect of jump discontinuity on speed of convergence. Integration and differentiation of Fourier series.
• Fourier transform, convolution, Dirac delta function. Laplace transform.
Course Prerequisites and Related Courses
|Students taking MP1 should have a good understanding of the material in Year 1 mathematics courses Calculus I&II and Linear Algebra I&II; or they can obtain the lecturer's permission.|
Course Evaluation Method and Criteria
|All students except Automotive Engineering Students:
Attendance and class participation: 5%; Weekly Quizzes and/or Assignments: 25%; Mid-term exam: 35%; Final Exam: 35%
Automotive Engineering Students:
Weekly Quizzes and/or Assignments, and Tutorial performance: 30%; Mid-term exam: 35%; Final Exam: 35%
|Boyce W., DiPrima R, Elementary Differential Equations, 7th Ed., Wiley.
You may use a newer edition, but note that the question numbers are different to those in the 7th edition.
|1. Boas M.L., 2006, Mathematical Methods in the Physical Sciences, 3rd ed., John Wiley & Sons.
2. Strang, G., Introduction to Linear Algebra, 4th Edition, Chapter 6.
3. Arfken G.B. & Weber H.J., 2005, Mathematical Methods for Physicists, 6th ed., Elsevier Academic Press.
(Copies of all these books are available in the Science Library.)
Study Load(Self-directed Learning Outside Course Hours)
|You are expected to revise the lecture notes, read the textbook, and solve assignment problems outside lecture hours. You cannot learn physics and mathematics by only attending lectures. The exams will consist of questions covering both lecture notes and assignments.|
Language Used in the Course
Lecture format, etc.
|Live lectures via MS Teams (Online only). Before the start of semester students should ensure that they have correctly installed MS Teams using their THERS (国立大学法人東海国立大学機構 ) email account.|
Additional measures for remote class (on-demand class)
|All lectures will be live via MS Teams (online only).
A lecture video will be available immediately after the lecture to help with student revision.
The lecturer will be available at most times during the day to answer questions via Teams chat.