Module also offered within study programmes:
General information:
Name:
Mathematical Analysis 2
Course of study:
2017/2018
Code:
IES-1-201-s
Faculty of:
Computer Science, Electronics and Telecommunications
Study level:
First-cycle studies
Specialty:
-
Field of study:
Electronics and Telecommunications
Semester:
2
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
dr Mc Inerney Kinga (stolot@agh.edu.pl)
Academic teachers:
dr Mc Inerney Kinga (stolot@agh.edu.pl)
Module summary

Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Social competence
M_K001 Knows how to explain mathematical phenomena in the understandable way ES1A_K06 Oral answer
Knowledge
M_W001 Is familiar with Fourier Series; knows how to expand a function into a Fourier Series ES1A_W01 Examination,
Test
M_W002 Is familiar with Fourier and Laplace Transforms ES1A_W01 Examination,
Test
M_W003 Has basic knowledge of Ordinary Differential Equations; knows how to solve non-homogeneous linear differential equation of higher order ES1A_W01 Examination,
Test
M_W004 Has knowledge of multi-variable differential calculus; knows how to find local and conditional extremas ES1A_K01 Examination,
Test
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Others
E-learning
Social competence
M_K001 Knows how to explain mathematical phenomena in the understandable way - + - - - - - - - - -
Knowledge
M_W001 Is familiar with Fourier Series; knows how to expand a function into a Fourier Series + + - - - - - - - - -
M_W002 Is familiar with Fourier and Laplace Transforms + + - - - - - - - - -
M_W003 Has basic knowledge of Ordinary Differential Equations; knows how to solve non-homogeneous linear differential equation of higher order + + - - - - - - - - -
M_W004 Has knowledge of multi-variable differential calculus; knows how to find local and conditional extremas + + - - - - - - - - -
Module content
Lectures:
Mathematical Analysis 2

The course covers 30h of lectures and 30h of classes.

LECTURES

1. The Fourier Serieses (4 h)
Trigonometric sequence as an orthogonal sequence. Trigonometric Fourier series. The Euler-Fourier Thm., Dirichlet’s Thm., Parsevall’s Condition, the Bessel inequality. Expanding a function into sine and cosine series. Complex Fourier series.
2. The Fourier intergral and the Fourier transform (2 h)
Fourier’s Thm., Sine and cosine Fourier transform. Complex Fourier transform and its properties.
3. The Laplace’s transform (4 h)
The Laplace transform and its properties. The inverse transform. Differentiating of the transform.
4. An Introduction to the Ordinary Differential Equations of the 1st order (1 h)
Definition and examples. General solution and Cauchy’s problem. Existance and uniquness of solutions.
5. Some methodes of solving different types of ODE’s (3 h)
Separation of variables (the Fourier method), linear differential equations of the 1st order, the Bernoulli equation.
6. Methods of solving nonhomogenious linear differential equation of higher order (3 h)
The method of variation of parameters (Lagrange’s method) – examples.
7. Functions of the multiple variables (3 h)
Neighbourhood of a point in R^n. Open, closed, bounded, compact and connected sets in R^n. The limit of a
sequence of points in R^n. The limit of a multivariable function, iterated limits. Continuous functions and their
properties (the Weierstrass Thm., the Darboux Thm.).
8. Derivatives of the multiple variable functions (4 h)
Directional derivative, partial derivative and their geometrical interpretation. Exact differential and its relation to directional derivatives and partial derivatives. Properties and the geometrical interpretation of the exact differential, and its matrix notation. The gradient of a function. Derivatives of vector value functions. Jacobi matrix. Differential of the composition of the functions.
9. Extrema of the multiple variable functions (4 h)
Higher order partial derivatives. Symmetry of second derivatives. Hessian matrix. Definition of the local maxima and the local minima. The necessary and sufficient conditions for the local extrema.
10. Vector fields (2 h)
Definition of a vector field. Potential of a vector field. Rotation. Divergence. Laplace operator.

Auditorium classes:
Mathematical Analysis 2

CLASSES:

1. Exanding functions into Fourier serieses, special cases of odd and even functions (6 h)
2. The Fourier and Laplace transform (2 h)
3. Solving differential equatios of the 1st order (4 h)
4. Solving linear differential equations of the higher order (4 h)
5. 1st class test (2 h)
6. Calculating limits of the multiple variable functions (2 h)
7. Differential calculus of the multiple variable functions (6 h)
8. Vector fields (2 h)
9. 2nd class test (2 h)

Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 116 h
Module ECTS credits 4 ECTS
Participation in lectures 28 h
Participation in auditorium classes 28 h
Realization of independently performed tasks 30 h
Preparation of a report, presentation, written work, etc. 30 h
Additional information
Method of calculating the final grade:

1. Warunkiem koniecznym uzyskania pozytywnej oceny końcowej OK jest otrzymanie pozytywnej oceny
z ćwiczeń i z egzaminu. Przy czym warunkiem dopuszczenia do egzaminu jest posiadanie oceny pozytywnej z ćwiczeń.
2. Po obliczeniu oceny średniej ważonej według wzoru SW = 0,49SOC+0,51SOE, gdzie SOC jest średnią arytmetyczną ocen uzyskanych we wszystkich terminach zaliczeń z ćwiczeń, a SOE jest średnią arytmetyczną ocen uzyskanych we wszystkich terminach z egzaminu, ocena końcowa OK jest obliczana według zależności:
if SW >4.75 then OK:=5.0 (bdb) else
if SW >4.25 then OK:=4.5 (db) else
if SW >3.75 then OK:=4.0 (db) else
if SW >3.25 then OK:=3.5 (dst) else OK:=3 (dst)

Prerequisites and additional requirements:

Wiedza z przedmiotów: Analiza matematyczna 1 i Algebra

Recommended literature and teaching resources:

1. J. Bird “Higher Engineering Mathematics”
2. B. Demidovitch “Problems in Mathematical Analysis”
3. W. F. Trench “Elementary Differential Equations”

Scientific publications of module course instructors related to the topic of the module:

Additional scientific publications not specified

Additional information:

None