MAT-60206 Mathematical Analysis, 5 cr
Lisätiedot
Suitable for postgraduate studies.
Vastuuhenkilö
Janne Kauhanen
Opetus
| Toteutuskerta | Periodi | Vastuuhenkilö | Suoritusvaatimukset |
| MAT-60206 2017-01 | 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 the sum, the product, the quotient, and the composition of continuous functions. Boundedness, the existence of min and max and the Intermediate Value theorem for continuous functions. | Uniform continuity. Monotonic functions and continuity of the inverse function. | |
| 3. | DIFFERENTIABILITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Linear approximation, the derivative of the sum, the product and the quotient of differentiable functions, the Chain Rule, extreme values, the Mean Value Theorem, and l'Hospitals Rule. | ||
| 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. | |
| 5. | SEQUENCES AND SERIES OF FUNCTIONS Pointwise and uniform convergence of sequences of functions. Properties preserved by uniform convergence: continuity, differentiability, integrability. | Pointwise and uniform convergence of series of functions. |
Oppimateriaali
| Tyyppi | Nimi | Tekijä | ISBN | URL | Lisätiedot | Tenttimateriaali |
| Book | Introduction to real analysis (Edition 2.04) | William Trench | Chapters 1-4 | No | ||
| Lecture slides | Mathematical Analysis | Janne Kauhanen | The lecture slides are in Moodle | No | ||
| Summary of lectures | Matemaattinen analyysi | Janne Kauhanen | Ladattavissa Moodlesta | No |
Esitietovaatimukset
| Opintojakso | P/S | Selite |
| MAT-01460 Matematiikka 4 | Mandatory | 1 |
| MAT-01466 Mathematics 4 | Mandatory | 1 |
1 . Either Matematiikka 1-4 or Mathematics 1-4
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 |