F2712 Mathematics 2

Faculty of Science
Spring 2011 - only for the accreditation
Extent and Intensity
4/3/0. 5 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).
Teacher(s)
Mgr. Michael Krbek, Ph.D. (lecturer)
Mgr. Pavla Musilová, Ph.D. (lecturer)
Mgr. Emília Kubalová, Ph.D. (seminar tutor)
Guaranteed by
prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Pavla Musilová, Ph.D.
Prerequisites (in Czech)
Středoškolská matematika, problematika předmětu Matematika I
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The discipline is a second part of Mathematics for students of bachelor studies of applied physics and non-physical programs. Its aim is to give students a knowledge and understanding of fundamental concepts of basic mathematical disciplines required for natural sciences and technical disciplines -- mathematical analysis, linear algebra and geometry, probability theory.

Absolving hthe discipline a student obtain following knowledge and skills:

* Understanding of the concept of linearity, ability of practical calculus in linear algebra and geometry (calculations with vectors and linear mappings in bases using matrix algebra, solving eigenvalue problem)
* Skills in calculations using curvilinear coordinates
* Solving simple differential equations and systems of differential equations, and their use for applications in physics, geometry, technical ddisciplines, chemistry, etc.
* Understanding of basic concepts of vector analysis and practical calculations including applications
Syllabus
  • 4.Linear algebra second time
  • 4.1 Vector spaces (1st week)
  • * groups, rings, fields
  • * finite-dimensional vector spaces: axioms, lienar dependent ind independent systems of vectors, bases, examples -- matrices as vectors
  • * reprezentation of vectors in bases
  • * vector subspaces, sum and intersection of subspaces, complements of subspaces, dimensions and bases of subspaces
  • 4.2 Linear mapping of vector spaces (2nd week)
  • * the concept of a line ar mapping, examples
  • * reprezentation of linear mappings in bases
  • * kernel and image of a linear mapping
  • * projections
  • 5. Coordinate systems
  • 5.1 Cartesian coordinate system (3th week)
  • * Cartesian coordinates in R2 a R3
  • * coordinate lines and planes
  • * element of a surface and a volume
  • 5.2 Curvilinear coordinates (3th a 4th week)
  • * partial derivatives
  • * polar and cylindrical coordinates, their coordinate lines and surfaces, elementary surface and elementary volume
  • * spherical coordinates, their coordinate lines and surfaces, elementary surface and elementary volume
  • * general curvilinear coordinates, their coordinate lines and surfaces, elementary surface and elementary volume
  • 6.Linear algebra last time
  • 6.1 Scalar product(5th a 6th week)
  • * scalar product
  • * orthonormal bases
  • * orthogonal projection, least squares method from the algebraical poit of view
  • 6.2 Eigenvalue problem (7th a 8th week)
  • * eigenvectors and eigenvalues of linear operators, diagonalization, spectrum
  • * orthogonal and symmetrical operators and their diagonal form
  • * linear operators and tensor quantities
  • * linearity in technical applications
  • 7.Ordinary differential equations
  • 7.1 First order equations (9th week)
  • * equations with separed variables, nuclear decay, absorprion of radiation, solution of equations
  • * linearity nad exponential laws
  • * linear equation
  • 7.2 Second order and higher order linear equations (9th a 10th week)
  • * homogeneous linear equation with constant coefficients
  • * inhomogeneous linear equation, solution by variation of constants method
  • * equations of motion for simple physical systems, oscillations
  • 7.3 Systems of linear differental equations (11th week)
  • * first order systems of equations
  • * second order systems of equations: oscillations of many body systems, examples
  • 8. A note on multiple variable functions
  • 8.1 Functions and their graphs (12th week)
  • * functions of two and three variables
  • * graphs of funcitons of two and three variables, quadratic surfaces
  • * partial derivatives, chain rule for composed functions
  • * total differential
  • * gradient
  • 8.2 Diferential operators (13th week)
  • * vector multiple variable functions, integral curves of vector fields
  • * divergence a rotation of a vector field, operator nabla and Laplace operator
Literature
  • http://physics.muni.cz/~pavla/teaching.php
  • KVASNICA, Jozef. Matematický aparát fyziky. Vyd. 2., opr. Praha: Academia, 1997, 383 s. ISBN 8020000887. info
Teaching methods
Lectures: theoretical explanation with practical examples
Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex probléme, homeworks, tests
Assessment methods
Teaching: lectures and exercises
Exam: written test (solving problems and test), oral exam
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.