F2712 Mathematics 2

Faculty of Science
Spring 2017
Extent and Intensity
4/3/0. 5 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).
Teacher(s)
Mgr. Pavla Musilová, Ph.D. (lecturer)
Mgr. Jan Benáček, Ph.D. (seminar tutor)
Mgr. Michal Pazderka, Ph.D. (seminar tutor)
Mgr. Michal Čech, Ph.D. (assistant)
Guaranteed by
prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Pavla Musilová, Ph.D.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Timetable
Mon 20. 2. to Mon 22. 5. Tue 9:00–10:50 F1 6/1014, Fri 8:00–9:50 F1 6/1014
  • Timetable of Seminar Groups:
F2712/01: Mon 20. 2. to Mon 22. 5. Wed 14:00–16:50 F3,03015, M. Pazderka
F2712/02: Mon 20. 2. to Mon 22. 5. Thu 16:00–18:50 F3,03015, J. Benáček
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 the 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 disciplines, 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, linear dependent and independent systems of vectors, bases, examples -- matrices as vectors
  • * representation 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 linear mapping, examples
  • * representation of linear mappings in bases
  • * kernel and image of a linear mapping
  • * projections
  • 5. Coordinate systems
  • 5.1 Cartesian coordinate system (3rd 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 separated variables, nuclear decay, absorption of radiation, solution of equations
  • * linearity and 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 differential 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 Differential 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
    required literature
  • MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika II pro porozumění i praxi (Mathematics II for understanding and praxis). první. Brno: VUTIUM (Vysoké učení technické v Brně), 2012, 697 pp. ISBN 978-80-214-4071-5. info
    recommended literature
  • KVASNICA, Jozef. Matematický aparát fyziky. Vyd. 2., opr. Praha: Academia, 1997, 383 s. ISBN 8020000887. info
  • MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika pro porozumění i praxi I (Mathematics for understanding and praxis I). Vydání druhé, doplněné. Brno: VUTIUM, VUT Brno, 2009, 339 pp. Vysokoškolské učebnice. ISBN 978-80-214-3631-2. info
Teaching methods
Lectures: theoretical explanation with practical examples
Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems, 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)
Study Materials
The course is taught annually.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for 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 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2017, recent)
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