Course Details

# Mathematics I

Academic Year 2022/23

GA01 course is part of 2 study plans

B-P-C-GK Winter Semester 1st year

B-K-C-GK Winter Semester 1st year

Linear algebra (basics of matrix calculus, rank of a matrix, solution of linear systems by Gauss elimination method). Inverse matrix, determinants. Eigenvalues and eigenvectors of a matrix.

Geometrical vectors in three dimensional Euclidean space, operations with vectors. Applications of vector calculus in spherical trigonometry. Vector space, base, dimension, coordinates of a vector. Application of vector calculus in analytic geometry.Real function of one real variable, limit and continuity of a function (basic notions and properties), derivative of a function (geometrical and physical meaning, techniques of differentiation, basic theorems on derivatives, higher order derivatives, sketching the graph of a function, differentials of a function, Taylor expansion of a function).

Course Guarantor

Institute

Objective

Being able to compute with matrices, performing elementary transactions, and calculating determinants, solving systems of linear algebraic equations by Gauss elimination method. Getting acquainted with the general properties of geometric vectors, without using coordinates. Knowing all about the dot, cross, and scalar triple products of geometric vectors, understanding their role in spherical trigonometry. Being able to apply such products when solving metric and positional problems in 3D analytic geometry. Understanding the basic ideas of the calculus of one- and two-functions including geometric interpretations of some concepts. Mastering differentiation, being able to sketch the graph of a function.

Knowledge

After passing the course students will have necessary skills for performing operations with vectors defined either generally or by their coordinates. Except this application of vectors in the spherical trigonometry will be deeply explained. The next outpiuts are: application of vector calculus in metrix and positional problems in analytical geometry, operations with matrices and solving systems of linear algebraic equations.

Bringing off basic differential calculus will permit successfully analyse problems of behavior of analytical curves.

Syllabus

1. Matrices, systems of linear algebraic equations, Gaussian elimination method.

2. Inverse matrix, determinants.

3. Geometrical vectors in three dimensional Euclidean space, operations with vectors.

4. Applications of vector calculus in spherical trigonometry.

5. Vector space, basis, dimension, coordinates of a vector.

6. Eigenvalues and eigenvectors of a matrix.

7. Application of vector calculus in analytic geometry.

8. Real function of one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic functions).

9. Polynomials and rational functions.

10. Sequences and their limits, limit and continuity of a function.

11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions.

12. Derivatives of higher order, geometrical meaning of first order and second order derivatives for sketching the graph of a function, l Hospital's rule, asymptotes.

13. Properties of functions continuous on an interval. Basic theorems of differential calculus (the Rolle and Lagrange theorems). Differential of a function. Taylor's theorem. Derivative of a function given in a parametric form.

Prerequisites

Basics of mathematics as taught at secondary schools. Graphs of elementary functions (powers and roots, quadratic function, direct and indirect proportion, absolute value, trigonometric functions) and basic properties of such functions. Simplifications of algebraic expressions.

Definition of a geometric vector and basics of 3D analytic geometry (parametric equations of a straight line, dot product of vectors and its applications to metric and positional problems). Identifying the the types and basic properties of conics, sketching graphs of conics)

Language of instruction

Czech

Credits

8 credits

Semester

winter

Forms and criteria of assessment

course-unit credit and examination

Specification of controlled instruction, the form of instruction, and the form of compensation of the absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Offered to foreign students

Not to offer

Course on BUT site

Lecture

13 weeks, 3 hours/week, elective

Syllabus

1. Matrices, systems of linear algebraic equations, Gaussian elimination method.

2. Inverse matrix, determinants.

3. Geometrical vectors in three dimensional Euclidean space, operations with vectors.

4. Applications of vector calculus in spherical trigonometry.

5. Vector space, basis, dimension, coordinates of a vector.

6. Eigenvalues and eigenvectors of a matrix.

7. Application of vector calculus in analytic geometry.

8. Real function of one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic functions).

9. Polynomials and rational functions.

10. Sequences and their limits, limit and continuity of a function.

11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions.

12. Derivatives of higher order, geometrical meaning of first order and second order derivatives for sketching the graph of a function, l Hospital's rule, asymptotes.

13. Properties of functions continuous on an interval. Basic theorems of differential calculus (the Rolle and Lagrange theorems). Differential of a function. Taylor's theorem. Derivative of a function given in a parametric form.

Exercise

13 weeks, 3 hours/week, compulsory

Syllabus

1. Geometrical vectors in E3, operations with vectors.

2. Applications of vector calculus in spherical trigonometry.

3. Vector space, base, dimension, coordinates of a vector.

4. Application of vector calculus in analytic geometry.

5. Matrices, systems of linear algebraic equations, Gaussian elimination method.

6. Inverse matrix, determinants.

7. Eigenvalues and eigenvectors of a matrix.

8. Real function of a one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite and inverse functions. Elementary functions.

9. Polynomials and rational functions.

10. Sequences and theirs limits, limit and continuity of a function.

11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions.

12. Derivatives of higher order, geometrical meaning of first and second order derivatives for investigation of behavior of a function, l`Hospitals rule, asymptotes.

13. Properties of function, continuous on an interval. Basic theorems of differential calculus. Differential of a function. Taylor’s theorem. Derivative of a function given in a parametric form. Primitive function, Newtons integral, its properties and computation. Riemann’s integral. Integration methods for indefinite and definite integrals.