### Mathematical Analysis 2

#### Course Description

Differential and integral calculus of several variables. Series and power series. Ordinary differential equations.

#### Learning Outcomes

1. Explain and relate basic results of differential calculus of several variables
2. Apply and interpret basic methods and skills of differential calculus of several variables
3. Demonstrate and apply basic skills of integral calculus of several variables
4. Explain the notion of convergence of series of numbers and functions and apply basic criteria for testing convergence
5. Demonstrate skills to solve basic types of ordinary differential equations
6. Create and solve mathematical model based on differential equations for engeneering problems
7. Show the ability for mathematical modelling and problem solving applying methods of mathematical analysis in engineering
8. Show the ability for mathematical expressing and logical reasoning

#### Forms of Teaching

Lectures

Lectures are held in two cycles, 6 hours per week.

Exercises

Excercises are held one hour per week.

Partial e-learning

Teaching materials and homeworks are accessible on course webpage.

Continuous Assessment Exam
Homeworks 50 % 0 % 0 % 0 %
Attendance 50 % 0 % 0 % 0 %
Mid Term Exam: Written 0 % 50 % 0 %
Final Exam: Written 0 % 50 %
Exam: Written 0 % 100 %
##### Comment:

Regular attendance of lectures and doing homework are conditions for taking exam.

#### Week by Week Schedule

1. Euclidean space R^n; Functions of several variables, Curves in R^n; Tangent line on the space curve; Vector functions; Derivative of vector function
2. Limit and continuity; Partial derivatives; Differential; Gradient; Tangent plane, Higher order derivatives; Schwartz theorem
3. Higher order derivatives; Schwartz theorem, Derivative of composite function and chain rule, Integrals depending on the parameter
4. Directional derivative; Derivative of implicit function; Theorem of implicit function, Second differential and quadratic forms, Taylor's formula
5. Extrema; Local extrema, Extrema of a function subject to constraints; Lagrange mutliplier, Least squares method
6. Double integral; Change of variables; Polar coordinates; Applications
7. Triple integral; Change of variables; Cylindrical and spherical coordinates; Applications
8. Midterm exam
9. Series of numbers; Convergence of series, necessary conditions, Series with positive terms; Criteria for convergence, comparison, D'Alambert's, Cauchy's, integral criterion, Series of real numbers, absolute, conditional and unconditional convergent series
10. Power series, area of convergence and radius of convergence, representation of a function, Taylor and Maclaurin series; Application of Taylor series, Convergence of function series; Uniform convergence; Differentiation and integration of function series
11. Notion of differential equation, the field of directions, orthogonal and izogonal trajectories, Equations with separated variables; Linear differential equation; Exact differential equation
12. Homogeneous equation; Bernoulli and Riccati equation, General first-order differential equations; Singular solutions, Numerical solving of differential equations; Euler's method; Taylor's method
13. Higher order differential equations; Decreasing the order, Linear differential equation of the second order; Homogeneous and nonhomogeneous equation, Examples; Harmonic motion; Applications in physics and electrical engineering
14. Higher order homogeneous equations, Finding the particular solutions, Solving equations using series
15. Final exam

(2. semester)

#### Literature

(.), A. Aglić Aljinović i ostali: Matematika 2, Element, Zagreb, 2016.,
(.), P. Javor: Matematička analiza 2, Element, Zagreb, 1999.,
(.), J. Stewart, Calculus Early Transcendentals, 9th Edition, Cengage Learning, 2020.,
(.), M. Pašić: Matematička analiza 2, Merkur ABD, 2004.,
(.), S. Lang: Calculus of Several Variables, Third Edition, Springer, 1987.,

#### General

ID 183361
Summer semester
7 ECTS
L1 English Level
L2 e-Learning
90 Lectures
0 Seminar
15 Exercises
0 Laboratory exercises
0 Project laboratory