Course: 101B Mathematical analysis B I

Lecturer: dr hab. Aleksander Strasburger

Semester: winter

Lecture hours per week: 4

Class hours per week: 4

Code: 11.101101B

Credits: 9

Syllabus:.

This first semester part of the three semester course belonging to the main stream of the study finishing with the MSc. degree aims at providing the student with the classical tools of calculus thus leading him to gain the ability of unaided solving of typical problems encountered in practice of a physicists (investigating functions of one and several variables, analysing convergence of sequences and series, both numerical and functional, computing simple integrals, solving differential equations etc) and a very important ability of formulating physical, and generally scientific issues in mathematical terms.

The necessary steps for developing mathematical intuition, e.g. by means of investigating solutions by qualitative methods, are provided along the course.

The stress is laid on discussing and explaining the basic notions, illustrating the uses and interpretations of fundamental theorems (examples and counterexamples) rather then on their actual proving.

It is not assumed on the part of the student, that he has done advanced level of maths and physics at school. However, it is assumed that the student is acquainted with elementary functions (polynomials, trigonometric and exponential functions, logarithms) and has acquired a certain experience in formulating and understanding abstract notions and logical inference.

The scope of the course can be described as a slightly extended (e.g. with elements of differential equations) programme of the advanced level of mathematics at the high school (profiled mathematical – physical classes).

The material covered is absolutely necessary for successful completing the next semesters of the course.

1. F. Leja, Rachunek różniczkowy i całkowy, PWN, Warszawa 1979 (i inne lata).
2. K. Kuratowski, Rachunek różniczkowy i całkowy, PWN, Warszawa, (wiele wydań).
3. W. Rudin, Podstawy analizy matematycznej, PWN, Warszawa, (wiele wydań).
4. K. Maurin, Analiza cz.1 - Elementy, PWN, Warszawa, (wiele wydań).
5. W.I. Arnold, Równania różniczkowe zwyczajne, PWN, Warszawa, 1975.
6. A. Sołtysiak, Analiza matematyczna, Cz. I i II, Wydawnictwo Naukowe UAM, Poznań, 1995.
7. R. Ingarden, L. Górniewicz, Analiza matematyczna dla fizyków, PWN, Warszawa, 1981.
8. K. Kuratowski, Wstęp do teorii mnogości i topologii, PWN, Warszawa, (wiele wydań).
9. W. Kleiner, Analiza matematyczna, (2 tomy), PWN, Warszawa, 1986-92.
10. Th. Bröcker, Analysis I, II (2 Auflage), Spektrum Akademischer Verlag, Heidelberg, 1995.
11. R. Strichartz, The Way of Analysis, Jones and Bartlett, Boston -London, 1995.

Prerequisites:

Pass-grade of class exercises, oral and written examination.

Course: 101C Mathematical analysis C I

Lecturer: prof. dr hab. Stanisław Woronowicz

Semester: winter

Lecture hours per week: 4

Class hours per week: 4

Code: 11.101101C

Credits: 9

The course is directed to those students that have predestination to abstract thinking and intend in the future to study theoretical physics in a more serious way.

1. Theory of real numbers. Axiomatic approach. Examples of argumentation where Archimedes axiom and continuity axiom are used. Bounded sets and their extremes.
2. Elements of topology: Metric spaces. Examples. Open balls. Neighbourhoods. Open subsets of a metric space. The characteristic properties of the class of all open sets. Closed subsets of a metric space. Examples. The characteristic properties of the class of all closed sets. Limits of sequences of points of a metric space. Uniqueness of the limit. The limit of a sequence of points of a closed set belongs to the set. Sequences with the Cauchy property. Complete metric spaces.
3. Topology of real line R: Intervals in R. Open and closed sets in R. Bounded increasing (decreasing) sequences of real numbers and their limits. Three sequences theorem. Completeness of R. Examples of complete spaces.
4. Topology (continuation): Continuity of a mapping in a point. Definition and characteristic properties. Continuous mappings. Examples. Composition of continuous mappings. Cartesian product of metric spaces. Subspaces of metric spaces. Continuity of the embedding. Cartesian product of continuous mapping. Continuity of arithmetic operations. The algebra of continuous functions. Sequential definition of compact spaces and compact sets. Properties of compact sets. Cartesian product of compact spaces. Open coverings of metric spaces and their compactness. Properties of continuous maps defined on compact spaces. Uniformly continuous mappings. Compact subsets of R and R^N. Connected sets in R. Connected sets and continuous maps. Darboux property of continuous functions.
5. Differential calculus of functions of one variable: The basic idea of differential calculus. Tangent functions of order n. Definition of the derivative. The derivative of the sum, product and quotient of differentiable functions. The derivative of the sum, product and quotient of differentiable functions. The derivative of polynomials and rational functions. The derivative of compositions of differentiable functions. Rolle theorem. Formulae of Lagrange and Cauchy. The derivative of the inverse function. Taylor formula and its applications. Extremes of differential functions. Rules of de l’Hospital. Convex functions and their properties. Examples.
6. Rieman integral: Directed sets and generalised sequences (nets). Definition of the Rieman integral. Examples. Integrability of a continuous function over a compact interval. Mean value theorem for integral calculus. Basic theorem coupling differential and integral calculus. Integration by parts and substitution.
7. Elementary functions: Logarithm. Definition and properties. Exponential function and its properties. Number e. Examples of sequences with limits expressed in terms containing number e. Taylor formula for exponential function. Convexity of exponential function and related inequalities. Series with positive entries and criterions of convergence. Series with complex entries. Convergent and absolutely convergent series. Addition and multiplication of series. Exponential function of complex variable. Definition and properties. Trigonometric functions. Definition and properties. Derivatives of trigonometric functions. Number pi. Definition. Formula e^{2p i}=1. Computation of p . Geometrical interpretation of trigonometric functions. Inverse trigonometric functions. Definitions and derivatives. Techniques of integration, in particular integration of rational functions, Euler substitutions and trigonometric substitutions.

1. Lecture notes
2. L. Schwartz, Kurs analizy matematycznej.
3. W. Rudin, Podstawy analizy matematycznej.
4. K. Maurin, Analiza cz.1- Elementy.
5. W. I. Arnold, Równania różniczkowe zwyczajne.
6. K. Kuratowski, Wstęp do teorii mnogości i topologii.

Prerequisites:

Pass-grade of class exercises, oral and written examination.

Course: 102A Physics A I – Mechanics

Lecturer: dr hab. Zygmunt Szefliński

Semester: winter

Lecture hours per week: 4

Class hours per week: 6

Code: 13.201102A

Credits: 12

This course is intended to give an introductory treatment of classical mechanics based on demonstration of experiments. The primary objectives are to provide the student with an understanding of mechanics as a quantitative science, based on observation and experiment and with an appreciation of the experimental laws and fundamental principles that describe the behaviour of physical world. The necessary mathematical techniques are introduced at the most propitious moments within the development of the central theme - physics. The course is elementary in that it deals with the basic elements of physics, a point worth recognising. In order to deepen the knowledge students solve the problems.

1. The rudiments of techniques of effective learning. Methods of the solution of problems in physics. How to pass examinations.
2. Scope of physics. Scale of physical quantities. Units of measurements. Fundamental units of space, time and mass. Measurements and its accuracy. Role of mathematics in physics. Vector and scalar quantities. Vector operations. Vectors and laws of physics. Fundamental interactions, interactions of elementary particles. Role of model in physics, examples.
3. Elements of static's. Rigid body. Centre of mass. Equilibrium of a rigid body.
4. Kinematics. Position, displacement, velocity and acceleration of the particle. Linear motion of a particle. Physical interpretation of the derivative. Motion in space. Angular co-ordinates.
5. Laws of dynamics. The inertia of matter. The concept of force. Inertial mass, weight and momentum. Conservation of momentum. Interaction between bodies. Friction, inertial forces. Collisions. Elastic and inelastic collisions.
6. Work and energy. Kinetic energy. Force and potential energy in conservative systems. The work-energy theorem. Conservation of energy in conservative systems. Power.
7. Central forces. Centripetal and centrifugal forces. Low of universal gravitation. Kepler’s laws of planetary motion. Solar system. Harmonic oscillator. Many body systems. Conservation laws.
8. Dynamics of the rigid body. Angular speed and acceleration. Energy. Moment of inertia. Angular momentum. Spinning top in simple approach. Rigid body pendulum. Elastic properties of the rigid body.
9. Galilean relativity. Inertial frame of reference. Noninertial systems. Galileo transformation. Motion in noninertial frame of reference. Inertial forces.
10. Special relativity. Speed of light as limit of velocity. Principle of relativity and Lorentz transformation. Time dilatation. Equivalence of mass and energy. Relativistic dynamics.
11. Photon. Photon as a zero mass particle. Photon energy. Photoelectric effect. Momentum of photon. Compton effect. Pressure of the radiation.

1. R. Resnick, D Halliday, Fizyka 1, PWN, 1996.

2. M.A.Herman, A. Kalestyński, L. Widomski, Podstawy Fizyki, PWN, 1997.

3. A.K. Wróblewski, J.A.Zakrzewski, Wstęp do Fizyki t. I, PWN, 1984.

4. J. Orear, Fizyka t. I , WNT.

5. C. Kittel, W.D. Knight, M.A. Ruderman, Mechanika, PWN; (Kurs Berklejowski).

6. R. Feynman, Wykłady z Fizyki t. I, PWN.

7. I.W.Sawiliew, Kurs Fizyki t. I, PWN.

1. A.Hennel, W. Krzyżanowski, W Szuszkiewicz, K.Wódkiewicz, Zadania i problemy z fizyki, PWN.

2. Problems in R. Resnick, D. Halliday, Fizyka 1.

3. Problems in J. Orear, Fizyka.

Written and oral examination.

Course: 102B Physics B I and 102C Physics C I – Mechanics

Lecturer: dr hab. Teresa Rząca-Urban

Semester: winter

Lecture hours per week: 4

Class hours per week: 4

Code: 13.201102BC

Credits: 12

The course is designed to give students a basic understanding of the key concepts of classical mechanics and special relativity, which will underpin many courses given in later years. It is also recommended for students intending to study other science s who wish to extend their knowledge of elementary physics. Lecture courses will be including many demonstrations to illustrate applications of some of the topics covered.

1. Scalars and vectors, frames of reference, velocity, acceleration, Galilean transformation.
2. Newton's three laws, inertial frames, inertial mass, the equations of motion.
3. Elementary applications of Newton's laws to static and dynamic problems, frictional forces, Hooke's law.
4. Simple harmonic motion- harmonic oscillator, damped oscillations, forced oscillations, resonance.
5. Circular motions, angular velocity vector.
6. The motion in noninertial reference frames, centrifugal force, Coriolis force, the Foucault pendulum, weather patterns.
7. Angular momentum, torque.
8. Work done by forces, potential and kinetic energies, power.
9. Constants of the motion, conservation of the linear momentum, angular momentum and energy.
10. Gravitational force, gravitational potential energy, motion under central forces, effective potentials, two-body problem, reduced mass.
11. Kepler's laws for planetary motion.
12. Elastic and inelastic collisions, centre of mass.
13. Rigid body motion, rotational motion (fixed axis), the moment of inertia, physical pendulum, the inertia tensor, precession (simple treatment).
14. Relativistic transformations of length and time, Lorentz transformation, twin paradox. Lorentz transformation of velocity, conservation of momentum, velocity-dependence of mass, conservation of energy for relativistic systems, rest energy of a particle, relation between total energy, mass and momentum of particle.
15. Some things never change-Lorentz invariants.

1. C. Kittel, W. D. Knight, M. A. Ruderman, Mechanika (t. I kursu Berkeleyowskiego).
2. A.K. Wróblewski, J. A. Zakrzewski, Wstęp do fizyki, t.I i t. II cz.l .
3. J. Orear, Fizyka, t. I i II.
4. W. Karaśkiewicz, Zarys teorii wektorów i tensorów, rozdz.1-3.
5. A. Hennel i inni, Zadania i problemy z fizyki, cz. I.

Prerequisites:

Two colloquia, written examination (test + problems).

Course: 103B Algebra and geometry B

Lecturer: dr hab. Piotr Podleś

Semester: winter and summer

Lecture hours per week: 2

Class hours per week: 2

Code: 11.101103B

Credits: 9

1. Complex numbers: basic properties, geometrical interpretation, de Moivre’s formula, n-th roots of complex numbers, cubic equations, the fundamental theorem of algebra
2. Vector spaces: systems of linear equations, subspaces, generators, linear independence, basis and dimension, the sum and intersection of subspaces
3. Linear mappings: matrices, kernel and image of a linear mapping, isomorphisms, the matrix associated with a linear map, rank of a matrix
4. Determinants: definition, the Laplace expansion, properties, Cramer’s formula, permutations, Cauchy Theorem, inverse of a matrix
5. Linear operators: eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton Theorem, decomposition into characteristic subspaces, diagonalizability, functions of linear operators, projections
6. Quadratic forms: the Lagrange method of diagonalization, Sylvester Theorem, finding of the signature, Gram-Schmidt orthogonalization
7. Inner product: properties, orthonormal bases, orthogonal complement, orthogonal projection, distance, selfadjoint and unitary operators, diagonalization of quadratic form in an orthonormal basis, quadric surfaces

1. P. Podleś, Algebra (lecture notes avaliable in the Faculty library).
2. S. Zakrzewski, Algebra i geometria
3. A. Białynicki-Birula, Algebra liniowa z geometrią
4. J. Komorowski, Od liczb zespolonych do tensorów, spinorów...

Prerequisites:

Class exercises are passed on the basis of colloquia and student’s activity, written and oral examination.

Course: 103C Algebra and geometry C

Lecturer: prof. dr hab. Kazimierz Napiórkowski

Semester: winter and summer

Lecture hours per week: 2

Class hours per week: 2

Code: 11.101103C

Credits: 9

1. Complex numbers, fields, polynomials. Cubic equations, the fundamental theorem of algebra and its consequences.
2. Permutations and groups.
3. Vector spaces.
4. Linear operators, systems of linear equations.
5. The conjugate space, dual pairs.
6. Multilinear mappings, determinant.
7. Trace, determinant and characteristic polynomial of a linear operator. Eigenvectors and eigenvalues, Jordan bases.
8. Quadratic forms. Invariants and diagonalization.
9. Euclidean and unitary spaces. Hermitian, unitary and normal operators.
10. Affine and Euclidean geometry. Linear and quadratic submanifolds.

Remark: The main feature of Algebra C is not a wider list of subjects in comparison to Algebra B, but a deeper and more complete treatment. The stress is put on the understanding of the role and mutual dependence of used notions.

S. Zakrzewski, Algebra i geometria, Z. G. Uniwersytetu Warszawskiego, 1994.

1. A.I. Kostrykin, J. I. Manin, Algebra liniowa i geometria.
2. A.I. Kostrykin, Wstęp do algebry.
3. P. Urbański, Wykład z algebry dla fizyków, Z. G. Uniwersytetu Warszawskiego, 1997.
4. J. Komorowski, Od liczb zespolonych do tensorów, spinorów, algebr Liego i kwadryk.
5. S. Lang, Algebra.
6. A.I. Kostrykin (red.), Zbór zadań z algebry.
7. M. Kordos, Wykłady z historii matematyki.

Prerequisites:

Pass of class exercises, oral and written examination.

Course: 104 Principles of experimental error analysis

Lecturer: dr hab. Teresa Tymieniecka

Semester: winter

2 hours/week in half a semester

1 hour/week

Code: 13.201104

Credits: 3

The course is some comprehensive introduction to statistical analysis of data and their graphical presentation as well as to fundamental idea of uncertainty in measurement. This is addressed to students without any experience in experimentation and in extracting information from data.

The workshop pedagogical approach is used. Firstly we make listeners realise that there are quantities in nature, which instead of one value can have a large variety of possible values often appearing with different probabilities; their values are provided either by nature or by act of measure. The students are guided to discover the main statistical concept: description of these quantities with one or two values and estimate of their precision. To explore statistical principle and to apply them the simplest statistical models are introduced (Gauss, Poisson, binomial) together with the simplest test based on the models (the 3sigma test, the chi-square test, the plus- minus test). Then the statistical interpretation of measurement is introduced together with rules of error propagation. Optimisation comes together with the least squares methods applied to linear problems.

The course is designed to foster active learning by minimising lectures and replacing them with hands-on activities. Students are expected to perform some home experiments which permit them to explore the meaning of concepts such as randomness, variability, sampling, confidence, tendency, significance and get familiar with experimental designing.

The ISO terminology is used.

1. Copies of transparencies from all the lectures will be available in the local library.
2. A.K.Wróblewski, J.A.Zakrzewski, Wstęp do Fizyki, rozdz. I.8-10 wraz z przypisem I.2-4.
3. G.L.Squires, Practical Physics.
4. H.Hänsel, Podstawy rachunku błędów.
5. John R.Taylor, An Introduction to Error Analysis.
6. H.Abramowicz, Jak analizować wyniki pomiarów?

Suggested: some basic knowledge of differential calculus.

A written colloquium (arithmetical problems) and a written project (a physical experiment designed and performed as a homework).

Course: 105A Mathematics A II

Lecturer: prof. dr hab. Witold Bardyszewski

Semester: summer

Lecture hours per week: 6

Class hours per week: 6

Code: 11.102105A

Credits: 15

Syllabus:

1. Vector spaces

• basis and dimensions
• linear transformations
• eigenvalues and eigenvectors

2. Ordinary differential equation

• integration of ODE's of first order
• linear equations of the second order
• systems of linear equations with constant coefficients

3. Differential calculus in vector space

• continuity of functions of n variables
• partial derivatives and differentiability
• extrema of functions of n variables
• inverse mapping and implicit functions

4. Analytic geometry

• parametric equations of curves and surfaces
• manifolds of rank 2

5. Fourier analysis

• Fourier series and Fourier integral
• Fourier's theorem

Literature:

1. G.M. Fichtencholz Rachunek Różniczkowy i Całkowy
2. A. Mostowski, M. Stark Elementy Algebry Wyższej
3. A. Mostowski, M. Stark Algebra liniowa

Mathematics IA.

Homework, colloquia, written examination.

Course: 105B Mathematical analysis B II

Lecturer: dr hab. Aleksander Strasburger

Semester: summer

Lecture hours per week: 4

Class hours per week: 4

Code: 11.102105B

Credits: 10

Many variable functions.

1. F. Leja - Rachunek różniczkowy i całkowy, PWN, Warszawa, 1979 (i inne lata).

1. K. Kuratowski, Rachunek różniczkowy i całkowy, PWN, Warszawa, (wiele wydań).
2. W. Rudin, Podstawy analizy matematycznej, PWN, Warszawa, (wiele wydań).
3. K. Maurin, Analiza cz.1 – Elementy, PWN, Warszawa, (wiele wydań).
4. W.I. Arnold, Równania różniczkowe zwyczajne, PWN, Warszawa, 1975.
5. A. Sołtysiak, Analiza matematyczna, Cz. I i II, Wydawnictwo Naukowe UAM, Poznań, 1995.
6. R. Ingarden, L. Górniewicz, Analiza matematyczna dla fizyków, PWN, Warszawa, 1981.
7. K. Kuratowski, Wstęp do teorii mnogości i topologii, PWN, Warszawa, (wiele wydań).
8. W. Kleiner, Analiza matematyczna, (2 tomy), PWN, Warszawa, 1986-92.
9. Th. Bröcker, Analysis I, II (2 Auflage), Spektrum Akademischer Verlag, Heidelberg, 1995.
10. R. Strichartz, The Way of Analysis, Jones and Bartlett, Boston -London, 1995.

Prerequisites:

Pass of class exercises, written and oral examination.

Course: 105C Mathematical analysis C II

Lecturer: prof. dr hab. Stanisław Woronowicz

Semester: summer

Lecture hours per week: 4

Class hours per week: 4

Code: 11.102105C

Credits: 10

1. Real vector spaces with a norm Banach spaces. Spaces of bounded linear maps and their norms. Finite-dimensional spaces. Continuity of linear maps defined of finite-dimensional spaces. Matrices as linear mapping of arithmetic spaces (R^N spaces). Completeness of finite-dimensional spaces. Adjoint spaces. Convex closed sets. Separation theorem (Hahn-Banach).
2. Complete metric spaces. Fixed point theorem of Banach. Applications.
3. First order differential calculus of functions of many variables. Differentiable mappings. The derivative of a mapping point (Frecht). Jacobi matrix and Jacobian. Composition of differentiable mappings and its derivative. Directional and partial derivatives. Gateau derivative. Mappings with continuous partial derivatives are differentiable. Mean value theorem. Local invertibility of differentiable mappings. Examples. Implicit function theorem. Examples.
4. Introduction to differential geometry: Loal co-ordinate systems. Transformations of local co-ordinates. k-dimensional surfaces (manifolds) in R^N. Surfaces given by a system of equations. Surfaces introduced by parametrization. Vectors and directional derivatives. Tangent vectors and tangent spaces to a k-dimensional surface in R^N.
5. Higher order of differential calculus: Multilinear maps. Derivatives of higher order. Partial derivatives of higher order. Commutativity of partial derivatives. Taylor formula for functions and mappings. Extremes and stationary points of of many variables. Methods of finding whether a given stationary point is a maximum, minimum or a saddle point. Constrained extrema. Lagrange multipliers.
6. Ordinary differential calculus: First order differential equations. Integrating factor. Elementary methods of solving for first order differential equations. Lowering of order for higher order differential equations. Theorem of existence and uniqueness for solutions of system of first order differential equations. Pointwise and uniform convergence of a sequence of functions. Completeness of space C[a,b]. Role of Lipshitz condition in the theorem on existence and uniqueness. Systems of homogeneous linear first order differential equations with constant coefficients. A resolvent of a system of homogeneous linear first order differential equations. Inhomogeneous linear equations. Dynamical systems. Existence and uniqueness theorem for equations of higher order. Linear equations of higher order.
7. Multidimensional Riemann integral: Cubs in R^N. Partitions of cubs. Riemann integral over cubs – definition. Integrability of piecewise continuous function. Zero measure sets. Smooth hypersurfaces as zero measure sets. Multidimensional and multiple integrals. Integrals over closed domains with piecewise smooth boundaries. Passing to multiple integrals – putting the correct limits for multiple integrals. Basic properties of integral: linearity and additivity with respect to the integration domain. Change of variable in multidimensional integrals. Differential forms and related operations.

1. Lecture notes.
2. L. Schwartz, Kurs analizy matematycznej.
3. W. Rudin, Podstawy analizy matematycznej.
4. K. Maurin, Analiza cz.1- Elementy.
5. W. I. Arnold, Równania różniczkowe zwyczajne.
6. K. Kuratowski, Wstęp do teorii mnogości i topologii.

Prerequisites:

Pass of class exercises. Written and oral examination.

Course: 106A Physics A II – Electricity and magnetism

Lecturer: dr hab. Jacek Ciborowski

Semester: summer

Lecture hours per week: 4

Class hours per week: 4

Code: 13.202106A

Credits: 10

1. Introduction to the field theory.
2. Scalar and vector potential.
3. Constant electric and magnetic fields. Polarisation and magnetisation. Boundary conditions.
4. Constant electric current, resistance, capacitance.
5. Electromotive force, Ohm’s law, Kirchhoff laws.
6. Currents in gases and liquids, laws of electrolysis.
7. Lorentz force, Ampere’s force.
8. Biot-Savart’s law.
9. Time dependent electric and magnetic fields; induction.
10. Maxwell equations, charge conservation.
11. Energy density of electromagnetic field.
12. Units.

1. A. K. Wróblewski i J. Zakrzewski, Fizyka, t.II cz.2.
2. R. Resnick i D. Halliday, Fizyka, t.II.
3. E. Purcell, Elektryczność i magnetyzm.
4. R. Feynmann, Feynmanna wykłady z fizyki, t.II cz.1.
5. S. Szczeniowski, Fizyka Doświadczalna, cz.3.
6. S. Frisz i A. Timoriewa, Kurs fizyki, t.II.

Physics I, Mathematics I

Two colloquia, written examination.

Course: 106B and 106C Physics B II and CII –Electromagnetism

Lecturer: prof. dr hab. Jan A. Gaj

Semester: summer

Lecture hours per week: 3

Class hours per week: 4

Code: 13.202106B

Credits: 10

Part A: "Kinematics" of fields and currents (description without rules of behaviour)

1. Electric field. Electric charge and intensity of electric field.

2. Differentiation and integration of fields: gradient, rotation, divergence, circulation, flux and their intuitive representation. Field lines.

3. Electric current and current density, microscopic picture. Conductors and insulators. Kirchhoff’s first rule and conservation of charge.

4. Magnetic field and magnetic moment, Lorentz force, Hall effect.

Part B "Dynamics" of fields and currents (fundamental laws of their behaviour)

1. Electrostatic field: its potential character, Gauss' law (integral and local form), vector D. Condenser and its capacity, depleted layer, field effect transistor. Coulomb’s law. Screening, the method of images. Energy in electric field.

2. Electric current: Ohm’s law and its local form, mobility and concentration of charge carriers, deviations from Ohm’s law. Microscopic picture of Ohm’s law, model of viscous force, relaxation time. Joule's heat. Sources of electric current: electromotive force and internal resistance. Optimal resistance of a load.

3. Electric circuits: charging of a capacitor through a resistor, integrating and differentiating circuits, second Kirchhoff’s rule, measurements of current and voltage, Wheatstone bridge, compensation measurements.

4. Alternating currents: intensity measurements, complex number formalism.

5. Magnetic field: Ampere's law – integral and local form, the law of Biot and Savart, vector H, examples. Absolute definition of ampere. Displacement current.

6. Electromagnetic induction: Faraday’s induction law, Lenz rule. Complete set of Maxwell equations in vacuum. Eddy currents, inductance and mutual inductance, circuits with inductance, energy in a coil. LC circuit, its oscillations and resonance. Tesla transformer.

Digression: Canonical ensemble

Thermal equilibrium, notion of temperature, empirical temperature. Probability distribution (discrete and continuous case), averaging. Ergodic hypothesis, canonical distribution.

Part C Influence of fields on matter

1. Dielectric polarisation: electric dipole moment, polarisation vector, polarizability, and susceptibility. Elastic and orientation polarisation mechanisms. Influence of geometry of the system, Clausius-Mossotti equation. Temporal dependence in po larisation: resonance, relaxation. Dielectric function, polarisation and conductivity. Plasma oscillations.

2. Magnetism of matter: dia-, para-, and ferromagnetism, magnetisation and susceptibility, phenomenological description of ferromagnetism. Solenoid with a core, transformer, DC generator and DC electric motor. Microscopic mechanisms in magnetism: diamagnetism, paramagnetism, mean field model.

3. Electric conduction in liquids and gases: electrolysis, galvanic cells, electric discharge in gases, neon lamp.

1. Lecture notes.
2. R. P. Feynman i in., Feynmana wykłady z fizyki.
3. A.Piekara Elektryczność, materia i promieniowanie.

A. Chełkowski Fizyka dielektryków.

Physics I

Pass of class exercises, written and oral examination.

Course: 107 Computer programming I (for students of physics)

Lecturer: MSc. Paweł Klimczewski

Semester: summer

Lecture hours per week: 2

Class hours per week: 2

Code: 11.001107

Credits: 4

1. The structure and function of computers. Common programmes on personal computers: operating system DOS and Norton Commander, word processors, viewers, compression, email, ftp; Windows, calculator, Word, Excel, Netscape etc.
2. Introduction to programming in C and/or C++: programme structure, blocks, variables, basic commands, functions, arrays, dynamic variables, recurrences, error detection.
3. Simple algorithms, their testing and optimisation.
4. Operating system UNIX compared to DOS, word processors, compilers, telnet, email, ftp, Netscape etc.

1. B. Stroustrup, Język C++.
2. S. B. Lippman, Podstawy języka C++.
3. T. L. Hansen, C++ zadania i odpowiedzi.
4. C. Delannoy, Ćwiczenia z języka C++.
5. P. Klimczewski, Skrypt, w przygotowaiu.

Prerequisites:

Pass of class exercises, examination.

Course: 108 Introduction to techniques of measurements and preliminary laboratory

Lecturer: dr hab. Tadeusz Stacewicz

Semester: summer

2 every two weeks

3 every two weeks

Code: 13.202108

Credits: 3

The lectures prepare students for exercises in Preliminary laboratory. The program includes elementary techniques of measurements of electrical signals. The use of voltmeters, ammeters and oscilloscopes is presented. Elementary laws of the electrical circuits, proper connecting of electric equipment and mutual interaction of the equipment with the analysed electric system is discussed. Finally, the physics of semiconductor devices such as diodes and transistors is presented.

Preliminary laboratory offers exercises with resistor and RLC circuits, diode rectifiers and transistor amplifiers. The schemes are build by students themselves. Experimental errors are discussed.

1. H. Abramowicz, Jak analizować wyniki pomiarów?
2. G. L. Squires, Praktyczna fizyka.
3. P. Horovitz, Sztuka elektroniki.
4. T. Stacewicz, A. Kotlicki, Elektronika w laboratorium naukowym.

Principles of experimental error analysis.

Examination:

Course: A101 Introduction to astronomy I (for students of astronomy)

Lecturer: dr Irena Semeniuk

Semester: winter

Lecture hours per week: 3

Class hours per week: 1

Code: 13.701A101

Credits: 2

1. Spherical co-ordinate systems. Basic formulae of spherical trigonometry.
2. The diurnal and yearly motion of the Sun. The seasons and Earth's climatic zones. The time and timekeeping in astronomy. The Earth's shape. Determination of geographical co-ordinates.
3. The Earth's atmosphere and magnetosphere. Atmospheric extinction and refraction.
4. The two-body problem. Kepler's laws. Elements of orbits. Perturbations. The observed motion of planets and Moon. Tides. Solar and Lunar eclipses. Precession and nutation. Aberration. Parallax and measurement of distances of celestial bodies.
5. Telescopes. Plate scale. Limiting magnitude. Focal ratio. Resolving power. Seeing.
6. Stellar photometry. Magnitudes. Photometric Systems. Stellar temperatures. Stellar classification. Luminosity classes. HR Diagram.

1. E. Rybka, Astronomia Ogólna.
2. J. Stodółkiewicz, Astrofizyka Ogólna z Elementami Geofizyki.
3. M. Kubiak, Gwiazdy i Materia Międzygwiazdowa.
4. M. Jaroszyński, Galaktyki i Budowa Wszechświata.
5. J. Mietelski, Astronomia w Geografii.
6. J. Kreiner, Astronomia z Astrofizyką.

Prerequisites: ---

Colloquia, written test and oral examination.

Course: A102 Introduction to astronomy II (for students of astronomy)

Lecturer: dr Irena Semeniuk

Semester: summer

Lecture hours per week: 3

Class hours per week: 1

Code: 13.702A102

Credits: 2

1. Proper motions. Tangential and radial velocities.
2. Determination of stellar masses. Mass function. Mass-Luminosity relation. Stellar radii measurements.
3. The Sun. The Sun's atmosphere. The solar wind. The active Sun.
4. The Galaxy. Its shape and rotation. The spiral arms. Interstellar matter.
5. Stellar clusters. Populations. Stellar structure and evolution. Energy sources. Advanced stages of stellar evolution. Planetary nebulae. White dwarfs. Neutron stars. Pulsars. Black holes.
6. Supernovae. Variable stars.
7. Classification of galaxies. Hubble's law. Cosmological models. The background radiation. Quasars. Gamma-Ray Bursts.

1. E. Rybka, Astronomia Ogólna.
2. J. Stodółkiewicz, Astrofizyka Ogólna z Elementami Geofizyki.
3. M. Kubiak, Gwiazdy i Materia Miedzygwiazdowa.
4. M. Jaroszyński, Galaktyki i Budowa Wszechświata.
5. J. Mietelski, Astronomia w Geografii.
6. J. Kreiner, Astronomia z Astrofizyką.

Prerequisites:

Colloquia, written test and oral examination.