授業の目的 【日本語】
Goals of the Course(JPN) | | | 専門基礎科目として数学及び物理学等を学んだ後，さらに進んで工学への応用を視野に入れてベクトル解析と偏微分方程式を修得します。力学や電磁気に関連する分野，物質や熱等の移動現象を伴う分野など工学の多くの問題には、座標変換、ベクトル場、線積分などベクトル解析の知識とその応用が必要になります。また、振動・波動、熱拡散等を定量的に扱う上で、微分方程式を使いこなせることが必要となります。そこで、この授業ではベクトル解析および常微分方程式について学び、演習を通じてそれらの知識を実際の工学上の問題に利用できるよう修得することを目的とします。 |
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授業の目的 【英語】
Goals of the Course | | 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. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | | 専門基礎科目として数学及び物理学等を学んだ後，さらに進んで工学への応用を視野に入れてベクトル解析と偏微分方程式を修得します。力学や電磁気に関連する分野，物質や熱等の移動現象を伴う分野など工学の多くの問題には、座標変換、ベクトル場、線積分などベクトル解析の知識とその応用が必要になります。また、振動・波動、熱拡散等を定量的に扱う上で、微分方程式を使いこなせることが必要となります。そこで、この授業ではベクトル解析および常微分方程式について学び、演習を通じてそれらの知識を実際の工学上の問題に利用できるよう修得することを目的とします。 |
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到達目標 【英語】
Objectives of the Course | | 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. |
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バックグラウンドとなる科目【日本語】
Prerequisite Subjects | | |
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バックグラウンドとなる科目【英語】
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 |
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授業の内容【日本語】
Course Content | | (１)ガイダンスおよびベクトル代数
(２)ベクトルの微分
(３)ベクトルの積分（１）：線積分、面積分
(４)ベクトルの積分（２）：ガウスの発散定理、グリーンの定理、ストークスの定理
(５)偏微分方程式の基本概念
(６)振動・波動のモデリング
(７)変数分離，フーリエ級数の利用
(８)熱拡散方程式の解法 |
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授業の内容【英語】
Course Content | | 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 |
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成績評価の方法と基準【日本語】
Course Evaluation Method and Criteria | | 出席（２０％）
毎回の小テスト（３０％）
期末試験（５０％）
以上の割合で、講義の目的が達成されたかを判断し、60％以上の達成を合格と
します。 |
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成績評価の方法と基準【英語】
Course Evaluation Method and Criteria | | 4th period
tutorial Attendance: 50%; Class performance: 50%
5th period
Attendance: 5%; Weekly Quizzes and Assignments: 25%; Mid-term exam: 35%; Final Exam: 35% |
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履修条件・注意事項【日本語】
Course Prerequisites / Notes | | |
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履修条件・注意事項【英語】
Course Prerequisites / Notes | | |
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教科書【日本語】
Textbook | | |
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教科書【英語】
Textbook | | | oyce W., DiPrima R, Elementary Differential Equations, 7th –10th Ed., Wiley. |
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参考書【日本語】
Reference Book | | |
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参考書【英語】
Reference Book | | 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.) |
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授業時間外学習の指示【日本語】
Self-directed Learning Outside Course Hours | | |
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授業時間外学習の指示【英語】
Self-directed Learning Outside Course Hours | | |
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使用言語【英語】
Language used | | |
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使用言語【日本語】
Language used | | |
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授業開講形態等【日本語】
Lecture format, etc. | | |
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授業開講形態等【英語】
Lecture format, etc. | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置【日本語】
Additional measures for remote class (on-demand class) | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置【英語】
Additional measures for remote class (on-demand class) | | |
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