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Course Catalog 2014-2015
MAT-60206 Mathematical Analysis, 5 cr |
Additional information
Suitable for postgraduate studies
Person responsible
Janne Kauhanen
Lessons
| Study type | P1 | P2 | P3 | P4 | Summer | Implementations | Lecture times and places |
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Requirements
Two mid-course exams or final exam. The grade of the course may be improved by bonus points collected from the weekly exercises.
Completion parts must belong to the same implementation
Learning Outcomes
This course introduces students to the fundamentals of mathematical analysis at an adequate level of rigor. Upon successful completion of the course, student will be able to: - Read mathematical texts and proofs, - Use the definitions and apply the basic results that are introduced during this course and - Produce rigorous proofs of results that arise in this course using direct and indirect proof, induction and epsilon-delta technique.
Content
| Content | Core content | Complementary knowledge | Specialist knowledge |
| 1. | The real numbers: the structure and properties of the real numbers, least upper bound, open and closed sets. | Limit points and the Bolzano-Weierstrass Theorem. | |
| 2. | The limit and continuity of functions defined on subsets of the real line: arithmetic operations and the Intermediate Value theorem. | Uniform continuity. | |
| 3. | Differentiability of functions defined on subsets of the real line: linear approximation, arithmetic operations, the Chain Rule, the Mean Value Theorem and l'Hospitals Rule. | Taylor's Theorem. | |
| 4. | The Riemann integral: the definition using Riemann sums and upper and lower sums, arithmetic operations, existence of the integral using upper and lower sums, integrability of continuous functions, the Mean Value Theorem for Integrals and the Fundamental Theorem of Calculus. Testing the convergence of improper integrals and conditional convergence. | ||
| 5. | Sequences: convergence, monotonic sequences, pointwise and uniform convergence of sequences of functions. | Caychy sequence. The effect of uniform convergence on the limit function with respect to continuity, differentiability and integrability. |
Instructions for students on how to achieve the learning outcomes
If the student is mastering the concepts, results and short proofs in concrete examples the evaluation is 3. For the grades 4 and 5 the student should in addition to the previous level be able to independently apply theory to deduce new results.
Assessment scale:
Numerical evaluation scale (1-5) will be used on the course
Partial passing:
Study material
| Type | Name | Author | ISBN | URL | Edition, availability, ... | Examination material | Language |
| Book | Introduction to real analysis (Edition 2.04) | William Trench | Chapters 1-4 | No | English | ||
| Summary of lectures | Matemaattinen analyysi | Janne Kauhanen | Ladattavissa Moodlesta | No | Suomi |
Prerequisites
| Course | Mandatory/Advisable | Description |
| MAT-01160 Matematiikka 1 | Mandatory | |
| MAT-01260 Matematiikka 2 | Mandatory | |
| MAT-01360 Matematiikka 3 | Mandatory | |
| MAT-01460 Matematiikka 4 | Mandatory |
Additional information about prerequisites
Vaihtoehtoisesti Laajan matematiikan opintojaksot tai hyvin suoritetut Insinöörimatematiikan opintojaksot (18-19 op).
Prerequisite relations (Requires logging in to POP)
Correspondence of content
| Course | Corresponds course | Description |
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More precise information per implementation
| Implementation | Description | Methods of instruction | Implementation |
| The course is lectured in english. |