Calculus II 1100-1AF21
The second semester of the mathematical analysis course (Calculus II) intended for physics students consists of two parts. The first part concerns differential calculus of functions of several real variables. The second part covers the theoretical foundations and solution techniques for ordinary differential equations.
Course outline:
1. Functions of several real variables: elements of topology of Rn continuity of multivariable functions, derivative, directional derivative, partial derivatives, Taylor’s formula.
2. Extrema of functions of several variables.
3. Important theorems: the inverse function theorem, the implicit function theorem, the rank theorem.
4. Elements of analysis on surfaces: description of curves and surfaces, tangent space, extrema of functions on surfaces (constrained extrema).
5. Integration of functions of several variables: the Riemann integral on Rn, the change of variables theorem, Fubini’s theorem, improper integrals and integrals depending on a parameter.
6. Ordinary differential equations: examples, elementary solution methods, the existence and uniqueness theorem for the Cauchy problem.
7. Linear differential equations: first-order linear equations, systems of first-order linear equations, higher-order linear equations with constant coefficients (homogeneous and nonhomogeneous), higher-order linear equations with variable coefficients, the Wronskian, Liouville’s theorem.
8. Selected types of second-order linear differential equations.
Description prepared by Katarzyna Grabowska, January 2009.
Main fields of studies for MISMaP
Course coordinators
Term 2025L: | Term 2024L: |
Mode
Prerequisites (description)
Learning outcomes
Student who has passed the exam should
- be able to use the differential calculus of functions of n real variables to dtermine properties such as continuity, differentiability, extremal values,
- use integral calculus as a tool for solving problems coming from physics such as finding moment of inertia,
- know theoretical background and technics of solving certain types of ordinary differential equations including different linear problems.
Assessment criteria
The final exam is divided into two parts: written and oral. It is necessary to pass both parts of the exam. The details change from year to year.
Practical placement
Not applicable
Bibliography
1. Walter Rudin, Principles of Mathematical Analysis
2. G. M. Fichtenholz, Differential and Integral Calculus, Vols. II and III
3. Paweł Urbański, Ananliza II (in Polish)
4. Andrzej Birkholc, Mathematical Analysis: Functions of Several Variables
5. V. I. Arnold, Ordinary Differential Equations
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes: