単位数: 2. 担当教員: 清水 幸弘. 開講年度: 2024. 科目ナンバリング: TEI-MAT302J.
○
Google Classroomのクラスコードは工学部Webページにて確認できます。
学部シラバス・時間割(https://www.eng.tohoku.ac.jp/edu/syllabus-ug.html)
1.目的
ラプラス変換,特殊関数,2階線形偏微分方程式について理解して,計算力を身につける.また,この科目を応用物理学のさまざまな分野に応用するための基礎を習得する.
2.概要
工学に現れる現象の解明に重要な役割をはたす応用数学の一部であるラプラス変換,2階線形微分方程式について,また工学に応用される特殊関数のうち,特にガンマ関数,ルジャンドル関数について,それらの基礎を学習する。
3.達成目標等
(1) ラプラス変換とその逆変換を理解し,計算ができるようになる.また,それらを微分方程式と積分方程式の解法に応用できる.
(2) ガンマ関数,ルジャンドル関数の定義や公式の導出を理解して,これらの特殊関数を使うことができる.
(3) 2階線形偏微分方程式の解法を学び,境界条件を満たす解を求めることができる.
講義は対面形式で実施する.お知らせなどにGoogle Classroom(クラスコード: 6jtatb4)を用いる.
Google Classroom class codes can be found on the School of Engineering website at https://www.eng.tohoku.ac.jp/edu/syllabus-ug.html
1. Objective
To understand Laplace transforms, special functions, and second-order linear partial differential equations, and to acquire calculation skills. To acquire the basis for applying this subject to various fields of applied physics.
2. Outline
Students learn the basics of Laplace transform and second-order linear differential equations, which are a part of applied mathematics that play an important role in elucidating phenomena that appear in engineering, as well as special functions applied to engineering, especially gamma function and Legendre function.
3. Objectives
(1) To understand Laplace transform and its inverse transform, and to be able to calculate them. Students will also be able to apply them to the solution of differential and integral equations.
(2) Understand the definitions and derivation of the formulas for the the gamma function and the Legendre function, and be able to use these special functions.
(3) Learn how to solve second-order linear partial differential equations and find solutions that satisfy the boundary conditions.
Lectures will be given in a face-to-face format. Google Classroom (class code: 6jtatb4) will be used for announcements, etc.
履修要望科目:全学教育科目の解析学A,B,C(常微分方程式),線形代数学A,B,数学物理学演習I,IIおよび工学部専門科目の応用数学A.
Particularly, students shuold attend to the class of Applied Mathematics A as the other strongly relevant subject because they need elementary knowledges of Fourier series, Fourier transformations and complex analysis to understand the present class, Moreover, Mathematical and Physical Practice I, II, Linear Algebra A, B and Analysis A, B, C including ordinary differential equations are relevant as other subjects
1.ラプラス積分とラプラス変換
2.ラプラス変換の性質
3.ラプラス逆変換1
4.ラプラス逆変換2
5.微分方程式と積分方程式へのラプラス変換の応用
6.ガンマ関数と解析接続
7.ガンンマ関数に関する公式
8.ルジャンドル関数
9.ルジャンドル関数に関する公式1
10.ルジャンドル関数に関する公式2
11.円柱座標または極座標を用いた変数分離法(3次元のラプラス方程式)
12.2次元のラプラス方程式と境界条件を満たす解
13.2次元のポアソン方程式と境界条件を満たす解
14.1次元の波動方程式
15. 期末試験
1. Laplace integral and Laplace transform
2. properties of Laplace transform
3. Laplace inverse transform 1
4. Laplace Inverse Transform 2
5. application of Laplace transform to differential and integral equations
6. gamma Function and analytic connection
7. formulas for gammafFunction
8. Legendre functions
9. formulas for Legendre functions 1
10. formulas for Legendre functions 2
11. variable separation method using cylindrical or polar coordinates (Laplace equation in three dimensions)
12. Laplace equation in two dimensions and solutions satisfying boundary conditions
13. Poisson's equation in two dimensions and Green's function
14. wave equation in one dimension
15. term-end examination
講義は教科書にそって黒板に計算を書いて実施する.講義ノートは黒板を写すのではなく,一緒に計算をしながら書く. 計算のフォローが間に合わなかった部分にマークを付け,次回の講義までに復習をする.復習のときに教科書やノートを参考にして,マーク部分の再計算を行う.
The lecture will be given by writing calculations on the blackboard according to the textbook.Lecture notes are not copied from the blackboard, but are written while performing calculations together.Mark the parts of the calculations that you did not follow up in time, and review them before the next lecture.During the review, students should recalculate the marked areas by referring to the textbook and notes.
期末試験(70%)と講義中に実施する数回の小テスト(30%)から評価する.
Evaluation will be based on the final exam (70%) and several quizzes given during the lecture (30%).
電子メールなどでアポイントメントを取ってから研究室に来室してください.
Students should visit my office after taking an appointment by e-mail.