Mathematics and Physics Seminar

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Code	Completion	Credits	Range	Language
17VSMF	Z	2	2S	Czech

Course guarantor:

Petr Písařík

Lecturer:

Petr Písařík, Milan Šiňor, Jana Urzová, David Vrba

Tutor:

Petr Písařík, Milan Šiňor, Jana Urzová, David Vrba

Supervisor:

Department of Natural Sciences

Synopsis:

The aim of the course is to broaden and deepen the mathematics and physics curriculum in relation to the course Mathematics and Physics for Laboratory Practice. The course focuses on practicing advanced concepts and their applications in engineering disciplines. Also included is a review of the basic principles of mathematics and physics, which will provide students with a solid foundation for understanding more complex topics. Emphasis is placed on linking theoretical knowledge with practical application in engineering practice. The seminar also serves as a support for successful completion of the final exam.

Requirements:

Credit is awarded for attendance (minimum attendance at 10 sessions).

Syllabus of lectures:

Syllabus of tutorials:

MATHEMATICAL PART

1. Numbers and functions: natural numbers, integers, real numbers, intervals, number systems, polynomials, functions of two or more variables, composite and inverse functions. Limits and continuity of functions. Number fields, basic operations with numbers and mathematical expressions, infinities, intervals and operations on them, powers and square roots, modification of expressions.

2. Equations, inequalities and their systems. Sequences, properties and types of sequences, limit of a sequence.

3. Counting with matrices: matrices and vectors, operations with matrices, commutative, associative and distributive laws, unit and zero matrices, transpose and inverse matrices.

4. Derivative of a function, derivative as rate of change, as directive tangent to a curve, derivative of a constant, linear and power functions, rules for calculating derivative of sum, difference, product and quotient functions, derivative of a composite function, derivative of functions of several variables, partial derivative. The progression of a function of one variable: definitional domain, local and absolute extremes, monotone functions, convexity, concavity and inflection points.

5. Fundamentals of integral calculus indefinite integral, table integrals, basic integration methods, definite integral, implicit integral; integral as a generalized sum, integral as area under a graph; properties of integral.

6. Functions of several variables. Local extrema, finding the bounded extremum, Lagrange multiplier.

7. Introduction to differential equations: definition of DR, types of DR, intuition, simple equations and models, direction field, general solution, solution with initial condition.

PHYSICAL PART

1. Thermodynamics, kinetic theory of substances - basic concepts, state quantities, length and volume thermal expansion, internal energy, calorimetric equation, thermodynamic laws, physics of the microworld, photoelectric effect, X-ray, laser, radionuclides. Statistical physics, Maxwell-Boltzmann distribution of velocities of molecules, Entropy, Enthalpy, ...

2. Oscillations and waves, harmonic motion, velocity and acceleration of oscillatory motion, forced oscillations - resonance, waves, types of waves, equations of successive waves. Electricity and magnetism: electric charge, Coulomb's law, electrostatic field, electric field in dielectrics and conductors, electric current, magnetostatic field and force, magnetic properties of a coil, magnetic properties of substances, Maxwell's equations in differential and integral form, physical interpretation of Maxwell's equations, relation of the speed of light in a vacuum to permittivity and permeability.

3. Light: wave and electromagnetic nature of light, diffraction, polarization, coherence, interference, reflection and refraction of light, fundamentals of geometrical optics, plane mirror, spherical mirror, lenses, optical instruments, eye as an optical system.

4. Atomic physics, origin and development of atomic theory, basic chemical laws, Dalton's atomic hypothesis, Thomson's model of the atom, discovery of the electron, elementary electric charge, Rutherford's model of the atom.

5. The hydrogen atom, quantum numbers describing the state of the electron in the atom, representation of atomic orbitals. Multi-electron atoms, Pauli exclusion principle, atomic shell construction, weak bond method, vector model of the atom, strong bond method.

6. Particles in external electromagnetic field, atom in electric and magnetic fields, Stark effect, Zeeman effect, magnetic moment of the atom,

7. Interaction of atoms, conditions for chemical bond formation, diatomic molecules, hydrogen molecule - clarification of homopolar covalent bonding. Multiatomic molecules, spectrum of molecules, vibration of molecules, rotation of molecules. Interaction of radiation with matter quantum mechanically (absorption, reflection and scattering)

Study Objective:

Study materials:

Required reading:

FEYNMAN, Richard Phillips, Robert B. LEIGHTON a Matthew L. SANDS. Feynmanovy přednášky z fyziky: revidované vydání s řešenými příklady. 3. vydání. Přeložil Ivan ŠTOLL. Praha: Fragment, 2019. ISBN 978-80-253-1642-9.

DELVENTHAL, Katka Maria, KISSNER, Alfred, KULICK, Malte: Kompendium matematiky. Knižní klub, 2017. ISBN 978-80-242-5420-3

REICHL, Jaroslav. Encyklopedie fyziky. http://fyzika.jreichl.com/.

Massachuttes Institute of Technology. http://ocw.mit.edu/courses/physics/.

OLŠÁK, Petr. Lineární algebra, učební text FEL ČVUT, http://petr.olsak.net/linal.html.