MAT-60206 Mathematical Analysis, 5 cr
Lisätiedot
Suitable for postgraduate studies.
Vastuuhenkilö
Janne Kauhanen
Opetus
| Toteutuskerta | Periodi | Vastuuhenkilö | Suoritusvaatimukset |
| MAT-60206 2016-01 | 3 |
Janne Kauhanen |
Final exam, weekly exercises and weekly written exercises. |
Osaamistavoitteet
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.
Sisältö
| Sisältö | Ydinsisältö | Täydentävä tietämys | Erityistietämys |
| 1. | THE REAL NUMBERS Field and ordering properties of the real numbers, supremum and infimum, the Completeness Axiom, limit points. | Open and closed sets. | Heine-Borel Theorem and Bolzano-Weierstrass Theorem. |
| 2. | THE LIMIT AND CONTINUITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Continuity of sum, product, quotient and composition of continuous functions. Boundedness, the existence of min and max and the Intermediate Value theorem for continuous functions. Monotonic functions and continuity of the inverse function. | Uniform continuity. | |
| 3. | DIFFERENTIABILITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Linear approximation, the derivative of sum, product and quotient of differentiable functions, the Chain Rule, extreme values, the Mean Value Theorem and l'Hospitals Rule. | Taylor's Theorem. | |
| 4. | THE RIEMANN INTEGRAL Riemann sums and upper and lower sums, existence of the integral using upper and lower sums (the Riemann Condition), integrability of monotone and continuous functions. Basic properties of the integral: linearity, comparison of integrals, the Triangle Inequality and additivity on intervals. The Mean Value Theorem, the Fundamental Theorem of Calculus. Local integrability and the improper integral, improper integrals of nonnegative functions, the Comparison Test, absolute and conditional convergence. | Integration by parts and the change of variable. | The set of discontinuities and integrability. |
| 5. | SEQUENCES AND SERIES OF FUNCTIONS Pointwise and uniform convergence of sequences and series of functions. | Caychy's uniform convergence criterion for sequences and series of functions, Weirstrass's M-Test, the effect of uniform convergence on the limit/sum function with respect to continuity, differentiability and integrability. |
Oppimateriaali
| Tyyppi | Nimi | Tekijä | ISBN | URL | Lisätiedot | Tenttimateriaali |
| Book | Introduction to real analysis (Edition 2.04) | William Trench | Chapters 1-4 | No | ||
| Summary of lectures | Matemaattinen analyysi | Janne Kauhanen | Ladattavissa Moodlesta | No |
Esitietovaatimukset
| Opintojakso | P/S | Selite |
| MAT-01160 Matematiikka 1 | Mandatory | |
| MAT-01260 Matematiikka 2 | Mandatory | |
| MAT-01360 Matematiikka 3 | Mandatory | |
| MAT-01460 Matematiikka 4 | Mandatory |
Tietoa esitietovaatimuksista
Alternatively Engineering mathematics (18-19 ECTS credits) with final grades 4-5.
Vastaavuudet
| Opintojakso | Vastaa opintojaksoa | Selite |
| MAT-60206 Mathematical Analysis, 5 cr | MAT-43650 Mathematical Analysis, 6 cr | |
| MAT-60206 Mathematical Analysis, 5 cr | MAT-60200 Mathematical Analysis, 5 cr |