Analysis II E
General data
Course ID: | 1100-1Ind05 |
Erasmus code / ISCED: |
11.102
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Course title: | Analysis II E |
Name in Polish: | Analiza II R |
Organizational unit: | Faculty of Physics |
Course groups: |
(in Polish) Fizyka, ścieżka indywidualna; przedmioty dla I roku Physics, individual path; 1st year courses |
ECTS credit allocation (and other scores): |
9.00
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Language: | Polish |
Main fields of studies for MISMaP: | astronomy |
Prerequisites (description): | Experience in elementary mathematical analysis (Analysis I R) and linear Algebra. |
Mode: | Classroom |
Short description: |
The second part of the course on mathematical analysis (extended level) addressed to students of physics. The course is devoted to the differential and integral calculus of several variables, ordinary differential equations and elements of general theory of integration. |
Full description: |
The aim of the course is to supply on extended level the necessary knowledge concerning differential and integral calculus of several variables, ordinary differential equations and elements of general theory of integration. Program: 1. Differential calculus of functions of several variables a) Continuity of many variable functions b) Norms in vector spaces c) Derivatives of functions of several variables (strong, directional, partial derivative) d) Derivation of a composite function e) Repeated differentiation, Taylor's formula f) Extreme values of functions of several variables g) The local inversion theorem h) Functions represented implicitly i) Surfaces, exttreme values on surfaces, Lagrange multipliers 2. Differential equations a) The existence and uniqueness of the Cauchy problem b) Linear differential equations c) Higher order differential equations d) Examples of imposing additional conditions for differential equations e) Equations (the systems of equations) with constant coefficients 3.Theory of integration a) Iterated integrals b) Fubini theorem c) Change of coordinates d) Sets of zero measures e) Integrals depending on parameters f) Improper integrals g) Differential forms, external derivative, Poincare Lemma h) Stokes theorem |
Bibliography: |
K. Maurin: Analisys P. Urbański, Analiza II i III |
Learning outcomes: |
1. Mastering the basics of mathematical analysis. 2. Gaining competence in reading and understanding mathematical texts. 3. Obtaining elementary techniques of analysis of functions of many variables. 4. Training in basic methods of solving the ordinary differential equations. 5. Obtaining skill and experience of identifying crucial mathematical properties of objects under study. |
Assessment methods and assessment criteria: |
Two mid - terms, final written exam, final oral exam. Grading criteria: mastering the subject, solving problems. |
Practical placement: |
none |
Classes in period "Summer semester 2023/24" (in progress)
Time span: | 2024-02-19 - 2024-06-16 |
Navigate to timetable
MO CW
TU WYK
W CW
TH FR CW
WYK
CW
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Type of class: |
Classes, 60 hours, 30 places
Lecture, 60 hours, 30 places
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Coordinators: | Piotr Sołtan | |
Group instructors: | Javier De Lucas Araujo, Marcin Kościelecki, Piotr Sołtan | |
Course homepage: | https://www.fuw.edu.pl/~psoltan/teach.php | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Examination |
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Requirements: | Algebra I E 1100-1Ind02 |
|
Mode: | Classroom |
|
Short description: |
(in Polish) Przestrzenie Banacha, analiza wielowymiarowa, równania różniczkowe, całki po iloczynach kartezjańskich |
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Full description: |
(in Polish) 1. przestrzenie Banacha (norma, operatory ograniczone, przestrzenie skończenie wymiarowe) 2. rachunek różniczkowy (pochodna odwzorowania, operatory odwracalne, reguła Leibnitza, pochodne kierunkowe, twierdzenie o wartości średniej, pochodne cząstkowe, odwzorowania wieloliniowe, wyższe pochodne, wzór Taylora, ekstrema) 3. twierdzenie o funkcji uwikłanej (twierdzenie Banacha o punkcie stałym, twierdzenie o lokalnej odwracalności, twierdzenie o funkcji uwikłanej, metoda mnożników Lagrange’a) 4. równania różniczkowe zwyczajne (całki z funkcji o wartościach wektorowych, zagadnienie początkowe, równania wyższych rzędów, istnienie i jednoznaczność rozwiązań, zależność od danych początkowych, równania liniowe) 5. całki po produktach przestrzeni (miara produktowa, twierdzenie Fubiniego, twierdzenie o zamianie zmiennych) 6. uzupełnienia (np. twierdzenia Stone'a-Weierstrassa) |
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Bibliography: |
(in Polish) K. Maurin - Analiza W. Rudin - Podstawy analizy matematycznej W. Rudin - Analiza rzeczywista i zespolona P. Urbański - Analiza dla studentów fizyki II |
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