## Mathematics I and Tutorial(4.0 credits) | |||||

Code | : | 10209 | |||

Course Type | : | Basic Specialized Courses | |||

Class Format | : | Lecture and Exercise | |||

Course Name | : | Fundamental and Applied Physics | Automotive Engineering | Automotive Engineering | |

Starts 1 | : | 2 Autumn Semester | 2 Autumn Semester | 2 Autumn Semester | |

Elective/Compulsory | : | Compulsory | Compulsory | Compulsory | |

Lecturer | : | John A. WOJDYLO Designated Professor | KITAHARA Teppei Designated Assistant Professor |

•Course Purpose |

5th period
This course is a companion course to Mathematical Physics II. This course introduces first order and second order ordinary differential equations and their solution methods. Students master analytical techniques for problems that arise in physics, engineering and chemistry. 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. Students will find this mathematical methods course helpful in other units such as Quantum Mechanics, Analytical Mechanics, Electricity and Magnetism, as well as in Automotive Engineering and other engineering courses. This course has dual aims: 1) to convey mathematical principles; 2) to improve students’ technical ability – i.e. ability to express intuition in mathematical terms and ability to solve problems. 4th period Students taking Mathematical Physics I should also take this tutorial class. This course introduces first order and second order ordinary differential equations and their solution methods. Students master exact and approximate analytical techniques for initial value problems that arise in physics, engineering and chemistry. Questions of existence, uniqueness and convergence are also discussed. Fourier series follow naturally from the 2nd order theory and these are investigated, too. |

•Prerequisite Subjects |

Prerequisites
Calculus I; Calculus II; Linear Algebra I; Linear Algebra II, or Consent of Instructor Related Courses Mathematical Physics Tutorial I, Mathematical Physics II |

•Course Topics |

Course Outline
• 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. It is desirable to read a textbook or reference materials before a class |

•Textbook |

oyce W., DiPrima R, Elementary Differential Equations, 7th –10th Ed., Wiley. |

•Additional Reading |

4th period
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 are available in the library.) 5th period 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 are available in the library.) |

•Grade Assessment |

4th period
tutorial Attendance: 50%; Class performance: 50% 5th period Attendance: 5%; Weekly Quizzes and Assignments: 25%; Mid-term exam: 35%; Final Exam: 35% |

•Notes |

•Contacting Faculty |

4th period
Office: BuES ilding, ES617 Email: abetomo@kmi.nagoya-u.ac.jp 5th period Office: Science Hall 5F 517 Phone: 052-789-2307 Email: john.wojdylo@s.phys.nagoya-u.ac.jp |

SyllabusSystem Ver 1.27a