• Semester 3, position 1

The objective or this course is to master basic methods for solving certain ordinary higher-order differential equations and systems of first-order differential equations, get acquainted with scalar and vector fields, and then to become able to compute multiple integrals and learn how to integrate scalar and vector fields over curves and surfaces in the space.

By the end of the course, the student is expected to be able to solve certain types of ordinary higher-order differential equations, as well as systems of first-order differential equations, to understand basic notions and properties of scalar and vector fields, to be able to compute multiple integrals both directly or using substitutions, as well as to understand and know how to compute integrals of scalar and vector fields over curves and surfaces in the space.

1. An introduction to higher-order differential equations, linearly independent solutions, reducing the order, autonomous equations. 2. Linear differential equation, reducing the order, homogeneous linear differential equation with constant coefficients, the characteristic polynomial, variation of constants, Euler's equation. 3. The method of elimination in systems of diff. eq., the eigenvalue method in systems with constant coefficients 4. Scalar and vector fields and the differential calculus with them, classification, field lines. 5. Explicit and parametric curves, integration of scalar or vector fields over a curve, path independence. 6. Double integrals, the order of integration, change of variables (particularly, polar coordinates), computing planar area, volume and surface area using double integrals. 7. Triple integrals, change of variables (particularly: cylinder and spherical coordinates). 8. Explicit and parametric surfaces, orientation of surface, integration of scalar or vector fields over a surface. 9. A glance over manifolds and differential forms, the theorems of Green, Gauss-Ostrogradsky and Stokes.

in accordance with the lectures

defined by the curriculum

"Differential equations" (in Serbian) - Faculty of Mechanical Engineering, Univ. of Belgrade, Department of mathematics (2017), ISBN: 978-86-7083-937-3 "Multiple, line and surface integrals" (in Serbian) - Faculty of Mechanical Engineering, Univ. of Belgrade, Department of mathematics (2023), YU ISBN: 978-86-6060-157-7 "Multivariable calculus", 7th edition - James Stewart, McMaster University and University of Toronto (2012), ISBN: 978-0-538-49787-9 Materials for lectures and tutorials (in Serbian): https://nastava.mas.bg.ac.rs/nastava/viewtopic.php?f=53&t=9607

Total assigned hours: 75

New material: 25
Elaboration and examples (recapitulation): 5

Auditory exercises: 30
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: 0
Test/test: 50
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 50
Requirement for taking the exam (required number of points): 25

"Differential equations" (in Serbian) - Faculty of Mechanical Engineering, Univ. of Belgrade, Department of mathematics (2017), ISBN: 978-86-7083-937-3; "Multiple, line and surface integrals" (in Serbian) - Faculty of Mechanical Engineering, Univ. of Belgrade, Department of mathematics (2023), YU ISBN: 978-86-6060-157-7; "Multivariable calculus", 7th edition - James Stewart, McMaster University and University of Toronto (2012), ISBN: 978-0-538-49787-9; Materials for lectures and tutorials (in Serbian): https://nastava.mas.bg.ac.rs/nastava/viewtopic.php?f=53&t=9607