# Mathematics III

2022/2023
Programme:
Physics, First Cycle
Orientation:
Educational Physics
Year:
2 year
Semester:
first
Kind:
mandatory
ECTS:
8
Language:
slovenian
Hours per week – 1. semester:
Lectures
5
Seminar
0
Tutorial
3
Lab
0
Prerequisites

Enrollment into the second academic year.

The prerequisite for the theoretical exam is a positive result of the written exam.

Homework may also be one of obligations.

Content (Syllabus outline)

Gamma and Beta special functions.

Double integral, triple integral and multiple integrals. Evaluation by iterated integrals. Center of mass, moment of inertia. Change of variables, polar, cylindric and spherical coordinates.

Inner product space, Hilbert space. Integrals of complex functions, space L_2[a,b]. Orthonormal systems, Fourier series, orthonormal basis (complete orthonormal system). Spaces L_2[-\pi,\pi]
and L_2[-a,a], pointwise convergence of trigonometric Fourier series.

Ordinary differential equations (DE): linear first order DE, separation of variables, exact DE. Existence theorem for the first order DE.

Existence theorem for a system of linear first order DE's. Homogeneous linear second order DE: Wronskian, linear DE with constant coefficients.
Nonhomogeneous linear second order DE: variation of parameters. DE with constant coefficients: particular solutions. Mechanical (damped and non-damped) oscillations, resonance. Linear DE's of higher order.

Systems of linear first order DE's. Systems with with constant coefficients: solving the homogeneus system using eigenvalues and eigenvectors. The exponential function of a square matrix, variation of parameters.

Line integrals, potential fields, Green's formula in the plane.

Surfaces in R^3, the tangent plane, area of a surface. Surface integrals and the flux of a vector field. Nabla (del) operator, gradient, divergence, curl. Gauss-Ostrogradski and Stokes' theorem. Operations with nabla, Laplacian (also in cylindric and spherical coordinates).

Calculus of variations: the Euler equation, isoperimetric problems, extrema with various constraints.

M. Dobovišek, Nekaj o diferencialnih enačbah, DMFA – založništvo, Ljubljana 2011.
E. Zakrajšek, Analiza III, Matematični rokopisi 21, DMFA-založništvo, Ljubljana 2002
A. Suhadolc, Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, Matematični rokopisi 23,  DMFA, Ljubljana, 1998.
Večina snovi je v (most of the course material is in):

Pri sestavljanju predavanj so bile uporabljene naslednje knjige (These books were used in compiling the course):
M. H. Protter, C. B. Morrey, Intermediate Calculus, 2nd edition, Undergraduate texts in Mathematics, Springer, New York, 1985.
J. E. Marsden, M. J. Hoffman, Elementary Classical Analysis, Freeman, San Francisco 1993.
A. Pinkus, S. Zafrany, Fourier Series and Integral Transforms, Cambridge University Press, Cambridge 1997.
G. Bachmann, L. Narici, E. Beckenstein: Fourier and wavelet analysis, Universitext, Springer-Verlag, New York 2000.
M. Braun, Differential Equations and Their Applications, 4th ed. Applied mathematical sciences 15, Springer-Verlag, New York 1993.
V. A. Zorich, Mathematical Analysis I and II, Universitext, Springer Verlag, Berlin Heidelberg 2004.
K. Jaenich, Analysis fuer Physiker und Ingenieure, Funktionentheorie, Differentialgleichungen, Spezielle funktionen, 3. Aufl., Springer Lehrbuch, Springer-Verlag, Berlin Heidelberg 1995.
L. Elsgolts, Differential equations and the calculus of variations, MIR Publishers, Moscow 1970.
S. Hassani, Mathematical Physics, A Modern Introduction to its Foundations, Springer-Verlag, New York 1999. (V poštev pride le majhen del te obsežne knjige - we need just a fraction of this book.)

Priročnik (Handbook):
E. Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, New York 2011

Vaje (problems and solved problems):
B. Hvala, Zbirka izpitnih nalog iz analize z namigi, nasveti in rezultati, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, 2000.
M. Dobovišek, Rešene naloge iz Analize II, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, 2001.
J. Cimprič, Rešene naloge iz Analize III, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, 2001.
M. Spiegel: Schaum's Outline of Advanced Mathematics for Engineers and Scientists
(Schaum's Outline Series), McGraw-Hill, New York 2009.
S. Lipschutz, D. Spellman, M. Spiegel: Vector Analysis and an introduction to Tensor Analysis, Second ed. (Schaum's Outline Series), McGraw-Hill, New York 2009.

Objectives and competences

Students learn advanced topics in Mathematical Analysis: multiple integrals, Fourier series, ordinary DE, vector analysis, calculus of variations. Mathematics III
is a basic course for physicists.

Intended learning outcomes

Knowledge and understanding:
We expect that students know important definitions and theorems, understand (and ideally be able to replicate) at least the easier proofs, and be able to apply this knowledge , e.g. in Mathematical Physics.

Application:
This course is a prerequiste for Mathematics IV, Numerical methods, Mathematical physics, Mechanics and other courses.

Reflection:
Students master some advanced topics in Mathematical Analysis and are able to apply them in physics.

Transferable skills:
Students learn to understand the usefulness of the abstract approach, are able to connect the acquired knowledge with what they already mastered. They also learn to use other written sources and the internet. They are able to identify and solve problems, hand in homework on time, and memorize the important topics.

Learning and teaching methods

Lectures, tutorials, homework (optional).

Assessment

Written exam or 2 midterm exams instead of the written exam
Oral exam or theoretical test, homework
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

prof. dr. P. Legiša:

• P. Legiša, Adjacency preserving mappings on real symmetric matrices. Math.
commun., Croat. Math. Soc., Divis. Osijek, 2011, vol. 16, no. 2, 419-432.

• P. Legiša, Automorphisms of Mn, partially ordered by the star order. Linear
multilinear algebra, 2006, vol. 54, no. 3, 157-188.

• P. Legiša, Automorphisms of Mn, partially ordered by rank subtractivity
ordering. Linear algebra appl. 2004, vol. 389, 147-158.
prof. dr. B. Magajna:

• B. Magajna, Sums of products of positive operators and spectra of Lüders
operators, Proc. Amer. Math. Soc. 141 (2013) 1349-1360.

• B. Magajna, Fixed points of normal completely positive maps on B(H), J. Math.
Anal. Appl. 389 (2012) 1291-1302.
81

• B. Magajna, The Haagerup norm on the tensor product of operator modules,
J. Funct. Anal. 129 (1995) 325-348.
prof. dr. J. Mrčun:

• I. Moerdijk, J. Mrčun: On the developability of Lie subalgebroids. Adv. Math.
210 (2007), 1-21.

• J. Mrčun: On isomorphisms of algebras of smooth functions. Proc. Amer. Math.
Soc. 133 (2005), 3109-3113.

• I. Moerdijk, J. Mrčun: On integrability of infinitesimal actions. Amer. J. Math.
124 (2002), 567-593.