Course title Mathematics 3 – Function of several variables
Code M103
Status Lectures(30), Exercises (30)
Level Basic course
Year 1. Semester 3.
ECTS 5 ECTS credits
Lecturer Tomislav Marošević, Associate Professor
Course objective The objective of the course is to provide insight into the fundamental parts of mathematics related to functions of several variables: the area of definition, continuity and limes, derivatives and integrals of functions of several variables. Students should be encouraged to think critically and to research.
Prerequisites Mathematics 1, Mathematics 2
Learning outcomes: After successfully completed course, students will be able to:

  1. recognize and explain the fundamental concepts of differential and integral calculus of real and vector functions of several variables, such as the continuity of functions, limits, partial derivatives and differential of function, as well as multiple, curve and surface integrals;
  2. to calculate partial derivatives of complex functions,  implicit and parametric functions;
  3. to use calculus to compute the tangent plane and normal vector, and to determine the local extremes of functions of several variables
  4. calculate areas and volumes using double and triple integrals;
  5. calculate curve and surface integrals, and use them to calculate lengths, areas and volumes.
Teaching activity ECTS Learning outcome Students activity Methods of evaluation Points
min max
Class attendance  1  20 Class attendance  Evidence list  10  20
Knowledge test (preliminary exam) 2 40 Preparation for written examination Written  preliminary exam  40  100
Final exam  2  40 Repetition of teaching materials Oral exam (and written exam) 40 100
Total 5  100  90  220
Consultations Consultations are held once a week.
Gained competencies In this course students are introduced to the differential and integral calculus of functions of several variables and vector functions. Primarily the focus is on situations in which the geometric view is possible, i.e. real functions of two or three variables, and functions from R to R2 and R3. During lectures, basic concept is introduced, analyzed, and illustrated by examples. During the exercises students train appropriate techniques to approach to specific problems and for solving them.
Content (Course curriculum) Real functions of several variables. Space Rn. Level-curves and level surfaces. Limits and continuity.

Partial derivatives and differentiability of functions of several variables. Partial derivatives of implicit functions and composite functions. Partial derivatives and differentials of higher orders.

Vector functions. Vector functions of one variable – the derivation and integration. Differentiability of vector functions of several variables; Jacobi matrix.

Applications of differential calculus of functions of several variables. Equation of tangent plane to the surface. Taylor’s formula. Extremes and conditional extremes.

Multiple integrals. Double integral – definition, properties, calculation, substitution variables (polar coordinates), applications. Triple integrals (cylindrical and spherical coordinates).

Line integrals (first and second). Concept, properties, calculation, applications.

Surface integrals (first and second). Concept, properties, calculation, applications.

Scalar and vector fields. Directional derivative of a scalar field. Gradient of a scalar field. The divergence of a vector field. Rotation of a vector field. Theorem of Gauss-Ostrogradsky. Stokes’ theorem.

Recommended reading
  1. S. Suljagić, Matematika II,
  2. Slapničar,  Matematika 2,
  3. S. Kurepa, Matematička analiza 3: Funkcije više varijabli, Tehnička knjiga, Zagreb, 1984.
  4. B.P. Demidovič, Zadači i upražnjenja po matematičeskomu analizu, FM Moskva, 1963.
Additional reading
  1. P. Javor, Matematička analiza 2, Element, Zagreb, 2000.
  2. Š. Ungar, Matematička analiza u Rn, Golden marketing-Tehnička knjiga, Zagreb, 2005.
  3. G.N. Berman, Zbornik zadač po kursu matematičesko analiza, Nauka, Moskva, 1972.
  4. S. Lang, Calculus of Several Variables, Springer, New York, 1987.
  5. M. Lovrić, Vector Calculus, Addison-Wesley Publ.\ Ltd., Don Mills, Ontario, 1997.
Instructional methods Lectures and exercises are mandatory.
Exam formats The exam consists of a written and oral part and it is taken after completion of lectures and exercises. During the semester, students can take two or more colloquiums that replace the written examination.
Language Croatian
Quality control and successfulness follow up An anonymous questionnaire
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