Mathematics 1
Code  Completion  Credits  Range  Language 

101MT01  Z,ZK  6  2P+3C 
 Lecturer:
 František Bubeník
 Tutor:
 Yuliya Namlyeyeva (guarantor), František Bubeník, Kateřina Janžurová
 Supervisor:
 Department of Mathematics
 Synopsis:

1. Sequences of real numbers, fundamental concepts and definitions, limits of sequences and methods for their calculating, the number e.
2. Functions of a real variable, fundamental concepts and definitions, limits (proper and improper) and methods for their calculating, continuity.
3. Basic theorems for continuous functions and their applications: Bolzano's and Weierstrass's theorems, derivatives and their geometric and physical meaning, derivative rules, derivative of composite and inverse functions.
4. Derivatives of higher orders, differentials of the 1st and higher orders, Lagrange's theorem and its consequences, l'Hospital's rules.
5. An analysis of functions sequent on the properties of the 1st and 2nd derivatives (intervals of monotony, local extremes, convexity and concavity, points of inflection, asymptotes).
6. Global (absolute) extremes on compact intervals, word problems. Taylor's theorem, Taylor's polynomial and its applications.
7. Vector (linear) spaces, the vector space of ordered ntuples, R2, R3, linear combinations, linear independence and dependence, bases, the dimension, subspaces.
8. Linear hull, matrices, the rank of a matrix, Gauss's algorithm.
9. Systems of linear algebraic equations, basic methods for solving, Gaussian elimination, Frobenius theorem.
10. Matrix multiplication, inverse matrices and their applications, matrix equations.
11. Determinants of the 2nd and 3rd orders, Sarrus's rule, inverse matrices by means of determinants, Cramer's rule.
12. Fundamental properties of geometric vectors. General form and parametric representation of a plane. Parametric equations of straight lines. A straight line as the intersection of two planes.
13. Relationship problems on straight lines and planes, deviations and distances of planes and straight lines. Application of analytic methods for solving geometric problems in the space.
 Requirements:
 Syllabus of lectures:

Differential Calculus of Functions of a Real Variable:
1. Sequences of real numbers, fundamental concepts and definitions, limits of sequences and methods for their calculating, the number e.
2. Functions of a real variable, fundamental concepts and definitions, limits (proper and improper) and methods for their calculating, continuity.
3. Basic theorems for continuous functions and their applications: Bolzano's and Weierstrass's theorems, derivatives and their geometric and physical meaning, derivative rules, derivative of composite and inverse functions.
4. Derivatives of higher orders, differentials of the 1st and higher orders, Lagrange's theorem and its consequences, l'Hospital's rules.
5. An analysis of functions sequent on the properties of the 1st and 2nd derivatives (intervals of monotony, local extremes, convexity and concavity, points of inflection, asymptotes).
6. Global (absolute) extremes on compact intervals, word problems. Taylor's theorem, Taylor's polynomial and its applications.
Linear Algebra:
7. Vector (linear) spaces, the vector space of ordered ntuples, R2, R3, linear combinations, linear independence and dependence, bases, the dimension, subspaces.
8. Linear hull, matrices, the rank of a matrix, Gauss's algorithm.
9. Systems of linear algebraic equations, basic methods for solving, Gaussian elimination, Frobenius theorem.
10. Matrix multiplication, inverse matrices and their applications, matrix equations.
11. Determinants of the 2nd and 3rd orders, Sarrus's rule, inverse matrices by means of determinants, Cramer's rule.
Analytic Geometry in Space:
12. Fundamental properties of geometric vectors. General form and parametric representation of a plane. Parametric equations of straight lines. A straight line as the intersection of two planes.
13. Relationship problems on straight lines and planes, deviations and distances of planes and straight lines. Application of analytic methods for solving geometric problems in the space.
 Syllabus of tutorials:

In general the exercises at each tutorial shall be based on the subject matter of the previous lecture. The exception is analytical geometry the last two weeks of the semester will be devoted to it which is not included in tutorials.
 Study Objective:

The course is based on the calculus learnt at school. It provides students with a fundamental knowledge of differential calculus, linear algebra and analytic geometry, by developing deeper understanding and extending the calculus skills.
 Study materials:

!Bubeník F.: Mathematics for Engineers, Prague, 2014, ISBN 9788001056202
!Bubeník F.: Problems to Mathematics for Engineers, Prague, 2014, ISBN 9788001056219
!Rektorys K.: Survey of Applicable Mathematics, Vol. I, II, ISBN 9401583080, 9789401583084
 Note:
 Timetable for winter semester 2019/2020:

06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon Tue Fri Thu Fri  Timetable for summer semester 2019/2020:
 Timetable is not available yet
 The course is a part of the following study plans:

 Building Structures (compulsory course)
 Building Structures (compulsory course)
 Building Structures (compulsory course)