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National Taiwan Normal University Course Outline Spring , 2027 |
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| I.Course information |
| Serial No. | 2260 | Course Level | Undergraduate / Master |
| Course Code | MAC0088 | Chinese Course Name | 實變分析（二） |
| Course Name | Real Analysis (II) | ||
| Department | Department of Mathematics | ||
| Two/one semester | 1 | Req. / Sel. | Sel. |
| Credits | 3.0 | Lecturing hours | Lecture hours: 3 |
| Teach in English | Y | Teach in National Languages | |
| Prerequisite Course | |||
| Comment | |||
| Course Description | |||
| Day & Class Period/Location | Mon. 8 Gongguan M310, Thur. 7-8 Gongguan M310 | ||
| Curriculum Goals | Corresponding to the Departmental Core Goal | ||
| 1. Cultivate Mathematics Professional Ability |
College: 1-1 Equipped with professional mathematics competences 2-1 Being able to communicate and express mathematically 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 Master: 1-1 Equipped with professional mathematics competences 2-1 Being able to communicate and express mathematically 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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| 2. Pathway to advanced analytics courses |
College: 1-2 Being able to reason and induct with mathematical logic 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 3-2 Possessing the abilities to think independently, criticize, and reflect 4-1 Being knowledgeable and being able to self-develop in the profession Master: 1-2 Being able to reason and induct with mathematical logic 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 3-2 Possessing the abilities to think independently, criticize, and reflect 4-1 Being knowledgeable and being able to self-develop in the profession |
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| 3. Raise the level of abstract thinking |
College: 1-3 Being able to think mathematically and critically 3-4 Having insights, intuitions, and senses of mathematics 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning Master: 1-3 Being able to think mathematically and critically 3-4 Having insights, intuitions, and senses of mathematics 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning |
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| 4. Interpret the connection between mathematics and other disciplines from a high perspective |
College: 1-5 Being able to use mathematics as tools to learn other subjects 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 2-4 Possessing the competences of lifelong learning 3-5 Having good taste for mathematics 4-4 Possessing global views from both scientific and humanistic perspectives, and being able to appreciate the values of other knowledge fields Master: 1-5 Being able to use mathematics as tools to learn other subjects 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 2-4 Possessing the competences of lifelong learning 3-5 Having good taste for mathematics 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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| II. General Syllabus |
| Instructor(s) | Daniel Spector/ 司靈得 | ||
| Schedule | |||
| 1. Electricity and Magnetism Heuristic idea and the derivation of Maxwell’s equations, The fundamental theorem of calculus, Green’s theorem, the divergence theorem, Vitali covering lemma, Hardy-Littlewood-Wiener maximal theorem, the Lebesgue differentiation theorem 2. PDE and well-posedness in the sense of Hadmard Existence, Uniqueness, and Continuous Dependence for Maxwell’s Equations 3. Measurements, Function Spaces, and Continuity Estimates for Maxwell’s Equations Essential supremum, Normed linear spaces, Banach Spaces, Lp-spaces, Separable spaces, Dual spaces, Holder inequality, Minkowski inequality, Lp Holder Estimate for Maxwell’s Equations, Holder-Holder Estimates, Lp-Lq Estimates, Chebychev’s Inequality and Weak-Type estimates 4.Weak Solutions and Well-posedness Revisited Test functions, integration by parts, Existence, Uniqueness and Continuous Dependence of Weak Solutions for Maxwell’s Equations, the Poisson Equation 5. Fourier Transform and Partial Differential Equations The Fourier transform, Fourier transform of a Gaussian, the heat equation, the solution of Poisson’s equation by integration of the heat kernel 6. Sobolev Inequalities The reinterpretation of continuity estimates as Sobolev inequalities, Sobolev inequalities by Hedberg’s estimate, Mironescu’s observation, and Fournier’s argument, the coarea formula and its connection with Sobolev inequalities |
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| Instructional Approach | |||
| Methods | Notes | ||
| Formal lecture | |||
| Group discussion | |||
| Grading assessment | |||
| Methods | Percentage | Notes | |
| Assignments | 40 % | ||
| Midterm Exam | 30 % | ||
| Final exam | 30 % | ||
| Adjustment methods for students | |||
| Required and Recommended Texts/Readings with References | D. Spector, A Potential Approach to Analysis, Course Notes. | ||