Mathematical Description of Natural Phenomena｜Kyoto University OpenCourseWare

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Mathematical Description of Natural Phenomena
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Numbering Code	U-LAS10 10013 LJ55	Year/Term	2022 ・ First semester
Number of Credits	2	Course Type	Lecture
Target Year	Mainly 1st year students	Target Student	For science students
Language	Japanese	Day/Period	Tue.4
Instructor name	KOHEI FUJITA (Graduate School of Engineering Associate Professor) IBA CHIEMI (Graduate School of Engineering Associate Professor)
Outline and Purpose of the Course	近年の高等学校の数学教育カリキュラム改訂に伴い，高校の数学と大学に入ってから学ぶ数学との間に以前より大きなギャップが生じている．そのため，工学で必要となる対象の把握やその根底にある原理の把握がより困難となってきている．微分方程式による自然現象の把握と解析などはその重要な一例である．このような事情を踏まえ，本科目ではまず高校の数学と大学の数学との間にある基本的な考え方や手法の差を埋めることを目的としていくつかの数学的概念の紹介を行い，さらに工学に現れる現象がいかに微分方程式を用いて有用に記述，解析され得るかを講述する．自然現象の例としては，ばねの単振動，建物の振動，流れの問題，熱伝導，波動などに関して詳しく述べる．講義を主体とするが，適宜，演習などを組み合わせて理解を深める． ※講義は原則日本語で行います。 -------------------------------------------------------------------------------------------------------------------- With the recent revision of the mathematics education curriculum in high　schools, a larger gap in mathematics than before has appeared between high　schools and universities. As a result, it is becoming more and more　difficult to grasp the object and the principles underlying it, which is　necessary in engineering. The understanding and analysis of natural　phenomena using differential equations is one of the most important　examples. Based on these circumstances, this lecture will first introduce some　mathematical concepts with the aim of bridging the gap in basic ideas and　methods between high school mathematics and university mathematics. In　addition, we will discuss how phenomena that appear in engineering can be　usefully described and analyzed using differential equations. 　 Examples of natural phenomena will be discussed in detail, including single vibration of springs, building vibration, flow problems, heat conduction, and waves. Lectures will be given as the main part of the course, and exercises will be combined as necessary to deepen understanding. ※In principal, this lecture will be given by Japanese.
Course Goals	高等学校で学んだ数学や物理の内容が自然現象を記述する上でどのように役立つかを理解することが可能となる．また，微分方程式が果たす役割を理解することが可能となる． -------------------------------------------------------------------------------------------------------------------- Students will be able to understand how the mathematics and physics content they learned in high school can be used to describe natural phenomena. It will also enable students to understand the role of differential equations.
Schedule and Contents	上記の目標を達成するため，以下の内容について講義する．１．集合と写像２．行列と線形変換３．微分とテイラー展開４．微積分の基本公式，２変数関数，多重積分，求積法５．複素数６．微分方程式と自然現象のモデル化７．微分方程式の解法具体的な授業計画は以下のとおりである．・集合と写像（1/2回：伊庭）　集合と写像について，その基本的考え方を解説する．・行列と線形変換（2回：伊庭）　行列の演算とその応用，平面の線形変換と行列，逆行列などを解説する．・微分とテイラー展開（1/2回：伊庭）　微分の考え方と微分方程式をたてる際に必要となるテイラー展開について解説する．・2変数（多変数）関数の増減と偏微分（1回：伊庭）　2変数（多変数）関数の増減，最大，最小について解説する．偏微分についても学ぶ．・微積分の基本公式，積分の変数変換（1回：伊庭）　積分の変数変換，多重積分など微積分の基本公式について解説する．・求積法（1回：伊庭）　楕円の面積，円周の長さ，円の面積，球の表面積，球の体積等から出発して，さまざまな形の面積，体積，表面積を求める方法について解説する．・複素数に慣れる（1回：伊庭）　三角関数と複素数，振動方程式と複素数など，複素数を用いて関数や方程式を表現することによって，より簡潔に統一的に現象を記述することができることを学ぶ．また，対数関数と自然対数の底 eについても解説する．・微分方程式と自然現象のモデル化(1回：藤田）　自然現象をモデル化し，数理的に表現する数学的手法として微分方程式が存在する．その入門的解説を行う．・微分方程式の立式（2回：藤田）　種々の例について微分方程式のたて方を解説する．対象として，ばねの単振動，建物の振動，流れの問題，熱伝導，波動などを扱う．・常微分方程式や偏微分方程式の解法（２回：藤田）・演習（2回：藤田）・期末試験／学習到達度の評価（１回）・フィードバック（１回） -------------------------------------------------------------------------------------------------------------------- In order to achieve the above goals, the following topics will be covered. 1. Mathematical set and mapping 2. Matrix and linear transformation 3. Differentiation and Taylor series expansion 4. Fundamental formula of calculus, 2-variable (multivariable) functions, multiple integral and quadrature 5. Complex numbers 6. Differential equations and modeling of natural phenomena 7. Solving differential equations [Details] ・Mathematical set and mapping（0.5 week: IBA）　Basic concept of set and mapping will be explained. ・Matrix and linear transformation（2 weeks: IBA）　Matrix operations and their applications, linear transformations of the plane and matrix, and inverse matrices will be explained. ・Differentiation and Taylor series expansion（0.5week: IBA）　The concept of differentiation and the Taylor series expansion required when formulating differential equations will be explained. ・Increase / decrease and partial differentiation of 2-variable (multivariable) functions (1 week: IBA) 　The increase / decrease, maximum, and minimum of 2-variable (multivariable) functions and partial differentiation are described. ・Fundamental formula of calculus, change of variables in Integrals（1 week: IBA）　The basic formulas of calculus such as multiple integral and change of variables in integral well be explained. ・Quadrature（1week: IBA）　We will explain how to obtain the area, volume, and surface area of various shapes, such as the area of the ellipse, the length of the circumference, the area of the circle, the surface area and the volume of the sphere. ・Introduction to complex numbers (1 week: IBA) the phenomena can be described more concisely and uniformly by expressing functions and equations using complex numbers such as trigonometric functions / oscillator equation equations. We also explain the logarithmic function and the base e of the natural logarithm. ・Differential equations, a mathematical method for the mathematical representation of natural phenomena, are explained. An introductory explanation of differential equations is given with specific examples of modeling natural phenomena using differential equations. (1 week: FUJITA) ・The physical meanings expressed by differential equations are explained, and the method of constructing differential equations is explained for various examples. Topics include single vibration of a spring, vibration of a building, flow problems, heat conduction, and waves. (2 weeks: FUJITA) ・Solving ordinary differential equations and partial differential equations will be explained. (2 weeks: FUJITA) ・Exercise （2 weeks: FUJITA) ・Examination（1 week）・Feedback（1 week）
Evaluation Methods and Policy	期末試験(80%)と課題レポートによる平常点(20%)を総合して評価する. -------------------------------------------------------------------------------------------------------------------- Evaluation will be based on the final examination (80%) and a report (20%).
Course Requirements	None
Study outside of Class (preparation and review)	講義プリントに記載された演習問題などを解くことが望ましい． -------------------------------------------------------------------------------------------------------------------- It is desirable to solve the exercises given in the lecture handouts.
References, etc.	-------------------------------------------------------------------------------------------------------------------- Introduced during class