Course detail
Mathematics I
FAST-GA01Acad. year: 2012/2013
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.
Linear algebra (basis of matric calculus, rank of a matrix, solution of linear systems by Gauss method of elimination). Inverse matrix, determinants. Eigenvalues and eigenvectors of a matrix.
Real function of a one real variable, limit and continuity of a function (basic notions and properties), derivative of a function (geometrical and physical meaning, calculus of derivation, basic theorems on derivatives, higher order derivatives, behavior of a function, differentials of a function, Taylor polynomials).
Primitive function, indefinite integrals, properties of indefinite integrals, basic indefinite integrals, methods of integration.
Language of instruction
Czech
Number of ECTS credits
8
Mode of study
Not applicable.
Guarantor
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Learning outcomes of the course unit
Operations with vectors defined either generally or by their coordinates.
Application of vectors in the spherical trigonometry.
Application of vector calculus in analytical geometry.
Operations with matrices and solving systems of linear algebraic equations.
Approximation of functions by Taylor polynomial.
Analysis of problems of behavior of analytical curves.
Application of vectors in the spherical trigonometry.
Application of vector calculus in analytical geometry.
Operations with matrices and solving systems of linear algebraic equations.
Approximation of functions by Taylor polynomial.
Analysis of problems of behavior of analytical curves.
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)
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)
Co-requisites
Not applicable.
Planned learning activities and teaching methods
Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations – lectures, seminars.
Assesment methods and criteria linked to learning outcomes
Abilities leading to successful solution of some typical classes of differential equations as well as necessary theoretical knowledge and its application will be positively estimated.
The final evaluation (examination) depends on assigned points (0-100 points), 30 points is maximum points which can be assigned during seminars. Final examination is in written form (estimated by 0-70 points ).
The final evaluation (examination) depends on assigned points (0-100 points), 30 points is maximum points which can be assigned during seminars. Final examination is in written form (estimated by 0-70 points ).
Course curriculum
1. Geometrical vectors in three dimensional Euclidean space, 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 fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic 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 order 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 (Rolles theorem, Lagranges theorem). Differential of a function. Taylors theorem. Derivative of a function given in a parametric form. Notion of a primitive function, Newtons integral, its properties and computation. Definition of Riemann integral. -
Integration methods for indefinite and definite integrals.
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 fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic 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 order 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 (Rolles theorem, Lagranges theorem). Differential of a function. Taylors theorem. Derivative of a function given in a parametric form. Notion of a primitive function, Newtons integral, its properties and computation. Definition of Riemann integral. -
Integration methods for indefinite and definite integrals.
Work placements
Not applicable.
Aims
After the course the students should be acquaited with the general properties of geometric vectors, know all about the dot, cross, and scalar triple products of geometric vectors, understanding their role in spherical trigonometry. Rhey should be able to apply such products when solving metric and positionalproblems in 3D analytic geometry.
They should be able to compute with matrices, perform elementary transactions, and calculate determinants, solve systems of linear algebraic equations by Gauss elimination method.
Pochopit základní pojmy diferenciálního počtu funkce jedné proměnné a geometrické interpretace některých pojmů. Zvládnout derivování a naučit se řešit úlohu průběhu funkce.
They should understand the principles of integration of elementary functions.
They should be able to compute with matrices, perform elementary transactions, and calculate determinants, solve systems of linear algebraic equations by Gauss elimination method.
Pochopit základní pojmy diferenciálního počtu funkce jedné proměnné a geometrické interpretace některých pojmů. Zvládnout derivování a naučit se řešit úlohu průběhu funkce.
They should understand the principles of integration of elementary functions.
Specification of controlled education, way of implementation and compensation for absences
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Recommended optional programme components
Not applicable.
Prerequisites and corequisites
Not applicable.
Basic literature
Dlouhý O., Tryhuk V.: Diferenciální počet I, Limita a spojitost funkce. FAST - studijní opora v intranetu, 2005. (CS)
Dlouhý, O., Tryhuk, V.: Diferenciální počet funkce jedné reálné proměnné. FAST, 2008. (CS)
Dlouhý O., Tryhuk V.: Diferenciální počet I, Derivace funkce. FAST - studijní opora v intranetu, 2005. (CS)
Larson R., Hostetler R.P., Edwards B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. (EN)
Novotný, J.: Základy lineární algebry. FAST - studijní opora v intranetu i tištěné texty, 2005. (CS)
Tryhuk, V., Dlouhý, O.: Vektorový počet a jeho aplikace. FAST - studijní opora v intranetu, 2005. (CS)
Dlouhý, O., Tryhuk, V.: Diferenciální počet funkce jedné reálné proměnné. FAST, 2008. (CS)
Dlouhý O., Tryhuk V.: Diferenciální počet I, Derivace funkce. FAST - studijní opora v intranetu, 2005. (CS)
Larson R., Hostetler R.P., Edwards B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. (EN)
Novotný, J.: Základy lineární algebry. FAST - studijní opora v intranetu i tištěné texty, 2005. (CS)
Tryhuk, V., Dlouhý, O.: Vektorový počet a jeho aplikace. FAST - studijní opora v intranetu, 2005. (CS)
Recommended reading
Not applicable.
Classification of course in study plans
Type of course unit
Lecture
39 hod., optionally
Teacher / Lecturer
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.
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.
Exercise
39 hod., compulsory
Teacher / Lecturer
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.
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.