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National Taiwan Normal University Course Outline Spring , 2027 |
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| I.Course information |
| Serial No. | 2308 | Course Level | Undergraduate |
| Course Code | MAU0163 | Chinese Course Name | 高等微積分（二） |
| Course Name | Advanced Calculus (II) | ||
| Department | Department of Mathematics | ||
| Two/one semester | 1 | Req. / Sel. | Req. |
| Credits | 4.0 | Lecturing hours | Lecture hours: 4 |
| Teach in English | Y | Teach in National Languages | |
| Prerequisite Course | Prerequisite course: 【MAU0162 Advanced Calculus (I)】 | ||
| Comment | |||
| Course Description | |||
| Day & Class Period/Location | Mon. 3-4 Gongguan M310, Thur. 6-7 Gongguan M417 | ||
| Curriculum Goals | Corresponding to the Departmental Core Goal | ||
| 1. To understand the foundation for the theory of calculus |
College: 1-1 Equipped with professional mathematics competences 1-5 Being able to use mathematics as tools to learn other subjects 2-2 Possessing the competences of transferring and contextualizing theories in mathematics and mathematics education 3-4 Having insights, intuitions, and senses of mathematics 4-1 Being knowledgeable and being able to self-develop in the profession 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning 4-4 Possessing global views from both scientific and humanistic perspectives, and being able to appreciate the values of other knowledge fields |
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| 2. To enhance the ability in thinking in abstract theory |
College: 1-3 Being able to think mathematically and critically 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 2-1 Being able to communicate and express mathematically 2-4 Possessing the competences of lifelong learning 3-2 Possessing the abilities to think independently, criticize, and reflect |
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| 3. To elevate the level of logical argumentation |
College: 1-2 Being able to reason and induct with mathematical logic 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity 4-2 Possessing a consistent and firm attitude toward pursuing truths |
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| II. General Syllabus |
| Instructor(s) | LIN, Yen-Chi/ 林延輯 | ||
| Schedule | |||
1. Euclidean spaces (3.5 weeks): Inner product structure on Euclidean spaces, metric space topology, connected sets, compact sets, Bolzano-Weierstrass theorem, Heine-Borel theorem. 2. Continuous functions on metric spaces (2.5 weeks): Comparison between definitions of continuous functions on topological spaces and metric spaces, connected sets and the intermediate value theorem, compact sets and the extreme value theorem, uniformly continuous functions, Arzèla-Ascoli theorem. 3. Differentiation on ¡n (5 weeks): Partial derivative, total differentiation as a linear transformation, chain rule, directional derivatives, higher derivatives, inverse function theorem, implicit function theorem, mean value theorem, Taylor series expansion, classification of critical points by Hessian, Lagrange multiplier method. 4. Integration on ¡n (4 weeks): Riemann integral on several variables, iterated integrals, Fubini theorem on continuous functions, change of variable formula, line integrals, Green’s formula. For a 16-week semester, 0.5 weeks are available to each of the midterm and the final exam. |
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| Instructional Approach | |||
| Methods | Notes | ||
| Formal lecture | Lectures on theories and examples | ||
| Problem-based learning | Having students to present proofs and solutions for real problems on board | ||
| Grading assessment | |||
| Methods | Percentage | Notes | |
| Assignments | 20 % | Weekly homework assignments | |
| Midterm Exam | 40 % | One midterm exam | |
| Final exam | 40 % | One final exam | |
| Adjustment methods for students | |||
| Required and Recommended Texts/Readings with References | 1. Wade, An Introduction to Analysis, 4th edition. 2. Rudin, Principles of Mathematical Analysis, 3rd edition. 3. Apostol, Mathematical Analysis, 2nd edition. 4. Pugh, Real Mathematical Analysis, 2nd edition. 5. Tao, Analysis I, 4th edition. |
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