日本語 English
| 開講年度/ Academic YearAcademic Year |
20232023 |
| 科目設置学部/ CollegeCollege |
理学部/College of ScienceCollege of Science |
| 科目コード等/ Course CodeCourse Code |
CA405/CA405CA405 |
| テーマ・サブタイトル等/ Theme・SubtitleTheme・Subtitle |
|
| 授業形態/ Class FormatClass Format |
対面（全回対面）/Face to face (all classes are face-to-face)Face to face (all classes are face-to-face) |
| 授業形態（補足事項）/ Class Format (Supplementary Items)Class Format (Supplementary Items) |
|
| 授業形式/ Class StyleCampus |
|
| 校地/ CampusCampus |
池袋/IkebukuroIkebukuro |
| 学期/ SemesterSemester |
春学期/Spring SemesterSpring Semester |
| 曜日時限・教室/ DayPeriod・RoomDayPeriod・Room |
月3･4406/Mon.3･4406 Mon.3･4406 |
| 単位/ CreditCredit |
22 |
| 科目ナンバリング/ Course NumberCourse Number |
MAT3210 |
| 使用言語/ LanguageLanguage |
日本語/JapaneseJapanese |
| 履修登録方法/ Class Registration MethodClass Registration Method |
科目コード登録/Course Code RegistrationCourse Code Registration |
| 配当年次/ Grade (Year) RequiredGrade (Year) Required |
配当年次は開講学部のR Guideに掲載している科目表で確認してください。配当年次は開講学部のR Guideに掲載している科目表で確認してください。 |
| 先修規定/ prerequisite regulationsprerequisite regulations |
|
| 他学部履修可否/ Acceptance of Other CollegesAcceptance of Other Colleges |
|
| 履修中止可否/ course cancellationcourse cancellation |
|
| オンライン授業60単位制限対象科目/ Online Classes Subject to 60-Credit Upper LimitOnline Classes Subject to 60-Credit Upper Limit |
|
| 学位授与方針との関連/ Relationship with Degree PolicyRelationship with Degree Policy |
|
| 備考/ NotesNotes |
|
| テキスト用コード/ Text CodeText Code |
CA405 |
Students will learn how to handle curves and surfaces accurately. In addition, we will learn about the curvature, which is an important invariant of these objects.
Curves and surfaces are common concepts in our daily lives, and we will use our knowledge of calculus to investigate them in detail. In the first part of the course, we will discuss curves, and learn what it means to treat them mathematically and rigorously. In addition, we will introduce the curvature, a basic invariant, and explain its geometric meaning.
Next, we deal with smooth surfaces (regular surfaces) and define the quantities called the first and second fundamental forms. After introducing principal curvature by geometric considerations, and thereby defining Gaussian and mean curvatures, we will see that these can be expressed in terms of the first and second fundamental forms. Then the relation between curvature and local shape of a surface is discussed, and an introductory exposition of minimal surfaces is given as an important example of surfaces. Finally, we will prove Gauss's fundamental theorem, which states that Gaussian curvature is actually determined only by the first fundamental form.
※Please refer to Japanese Page for details including evaluations, textbooks and others.
曲線や曲面を正確に扱う方法を習得する。さらに，図形の重要な不変量である曲率について詳しく学ぶ。
Students will learn how to handle curves and surfaces accurately. In addition, we will learn about the curvature, which is an important invariant of these objects.
曲線や曲面は日常生活でもよく用いられる概念であるが，それらを詳しく調べるために微積分の知識を活用する。はじめに曲線について取り上げ，図形を数学的に厳密に扱うとはどういうことか習得する。さらに，基本的な不変量である曲率を導入し，その幾何学的な意味を説明する。
次に滑らかな曲面（正則曲面）について扱い，まず第一・第二基本形式と呼ばれる量を定義する。幾何学的考察により主曲率を導入し，それによりガウス曲率および平均曲率を定義した後，これらが第一・第二基本形式により表されることを見る。その後曲率と曲面の局所的な形状の関係について述べ，曲面の重要な例として極小曲面の入門的解説をする。最後に，ガウス曲率が実は第一基本形式のみで決まるという「ガウスの基本定理」を証明する。
Curves and surfaces are common concepts in our daily lives, and we will use our knowledge of calculus to investigate them in detail. In the first part of the course, we will discuss curves, and learn what it means to treat them mathematically and rigorously. In addition, we will introduce the curvature, a basic invariant, and explain its geometric meaning.
Next, we deal with smooth surfaces (regular surfaces) and define the quantities called the first and second fundamental forms. After introducing principal curvature by geometric considerations, and thereby defining Gaussian and mean curvatures, we will see that these can be expressed in terms of the first and second fundamental forms. Then the relation between curvature and local shape of a surface is discussed, and an introductory exposition of minimal surfaces is given as an important example of surfaces. Finally, we will prove Gauss's fundamental theorem, which states that Gaussian curvature is actually determined only by the first fundamental form.
| 1 | 準備（ベクトル積と常微分方程式） |
| 2 | 平面曲線とその曲率 |
| 3 | 平面曲線の基本定理 |
| 4 | 空間曲線の曲率と捩率 |
| 5 | 空間曲線の基本定理 |
| 6 | 正則曲面の定義と接空間 |
| 7 | 法ベクトルとガウス写像 |
| 8 | 曲面の第一基本形式 |
| 9 | 曲面の第二基本形式 |
| 10 | 主曲率とガウス曲率および平均曲率 |
| 11 | 曲率と曲面の局所的形状 |
| 12 | 極小曲面 |
| 13 | ガウスの公式とワインガルテンの公式 |
| 14 | ガウスの基本定理 |
板書 /Writing on the Board
スライド（パワーポイント等）の使用 /Slides (PowerPoint, etc.)
上記以外の視聴覚教材の使用 /Audiovisual Materials Other than Those Listed Above
個人発表 /Individual Presentations
グループ発表 /Group Presentations
ディスカッション・ディベート /Discussion/Debate
実技・実習・実験 /Practicum/Experiments/Practical Training
学内の教室外施設の利用 /Use of On-Campus Facilities Outside the Classroom
校外実習・フィールドワーク /Field Work
上記いずれも用いない予定 /None of the above
２変数の微分と積分（特に微分）について理解していることを前提とする。
| 種類 (Kind) | 割合 (%) | 基準 (Criteria) |
|---|---|---|
| 筆記試験 (Written Exam) | 50 | |
| 平常点 (In-class Points) | 50 |
授業内課題(20%) 小テスト(30%) |
| 備考 (Notes) | ||
| 「幾何学1演習」と一体で評価する。 | ||
なし/None