Advanced course in mathematics

Introduction to the linear algebra, functional analysis, analytical theory of partial differential equations and their most important applications in engineering.

Upon successful completion of this course, students should be able to: 1) apply learned methods of linear algebra and functional analysis to slove technical problems; 2) determine analitical solution of partial differential equation of first order; 3) determine analitical solution of basic problems of mathematical physics.

Distance, norm, inner product. Banach's fixed point principle and its applications. Orthonormal sets of vectors. Gram-Schmidt method. Linear operators. Characteristic and minimal polynomial of a square matrix. Reduction of a square form to a canonical form. Matrix analysis. Symmetric form of systems 1st order differential equations. Pfaffs equation. 1st order partial differential equations. Quasilinear equation: general, singular and particular solution. Nonlineaar equations: general, singular and particular solution. 2nd order partal differential equations - classification of equations of mathematical physics. Fourier method. D'Alambert method for hyperbolic equations. Goursat problem for hyperbolic equations. Laplace transform.

Distance, norm, inner product. Orthonormal sets of vectors. Banach's fixed point principle and its applications. Gram-Schmidt method. Linear operators. Characteristic and minimal polynomial of a square matrix. Reduction of a square form to a canonical form. Matrix analysis. Symmetric form of systems 1st order differential equations. Pfaffs equation. Systems of 1st order partial differential equations. 2nd order partal differential equations - classification of equations of mathematical physics. 2nd order partal differential equations - classification of equations of mathematical physics. Fourier method. D'Alambert method for hyperbolic equations. Goursat problem for hyperbolic equations. Laplace transform.

No prerequisites.

1. D. Georgijević, Partial differential equations (in Serbian), Faculty of Mechanical Engineering, Beograd 2002. 2. Z. Mamuzić, Selected topics from the theory of ordinary and partial differential equations, Faculty of Mechanical Engineering (in Serbian), Faculty of Mechanical Engineering, Beograd 1981. 3. D. S. Mitrinović, J. D. Kečkić, Equations of mathematical physics (in Serbian), Građevinska knjiga, Beograd, 1985. 4. S. Nešić, Matrices, Faculty of Mechanical Engineering (in Serbian), Faculty of Mechanical Engineering, Beograd 1992. 5. J. Kečkić, Linear algebra (in Serbian), Naučna knjiga, Beograd, 1990.

Total assigned hours: 65

New material: 40
Elaboration and examples (recapitulation): 10

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 10
Review and grading of lab reports: 0
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 0
Test: 0
Final exam: 5

Activity during lectures: 5
Test/test: 0
Laboratory practice: 0
Calculation tasks: 45
Seminar paper: 0
Project: 0
Final exam: 50
Requirement for taking the exam (required number of points): 20

M. Spalević, I. Aranđelović, G. Lazović, A. Pejčev, D.Đukić, D. Jandrlić, J. Tomanović, R. Mutavidžić - Đukić, Ordinary differential equations and series; Lj. Ćirić, Special functions, Faculty of Mechanical Engineering (in Serbian), Beograd 1984.; B. Đerasimović, Functions of complex variable. Euler's integrals, Faculty of Mechanical Engineering, Beograd 1969.; D. S. Mitrinović, P. M. Vasić, Differential equations - Collection of exercises and problems (in Serbian), Naučna knjiga, Beograd 1990.; S. Radenović, Linearna algebra, Methodological collection of exercises collection of problems (in Serbian), Naša knjiga, Beograd, 2007