Engineering Mathematics I
| Type of study | Bachelor |
|---|---|
| Language of instruction | Czech |
| Code | 310-3101/03 |
| Abbreviation | DPI |
| Course title | Engineering Mathematics I |
| Credits | 8 |
| Coordinating department | Department of Mathematics and Descriptive Geometry |
| Course coordinator | RNDr. Jan Kotůlek, Ph.D. |
Subject syllabus
1. The language of mathematics; basic concepts of mathematical logic and linear algebra; solving systems of linear equations. The Gaussian elimination method.
2. Matrix operations; rank of a matrix; the Frobenius theorem. Computation of the inverse matrix using the Gaussian method. Matrix equations.
3. Determinants; properties of determinants; computation of determinants of degree 2 and 3; Cramer's rule.
4. Real-valued functions of a single real variable: definition, graphs. Properties of real functions; operations with functions; composition of functions; inverse functions.
5. Introduction to limits. Sequences and their limits. Definition of the limit of a function. Continuous and discontinuous functions and their properties. Theorems on continuity.
6. Derivative of a function; its geometric and physical meaning. Differentiation rules for power functions. Linearity of the derivative; derivatives of products, quotients, and composite functions. Derivatives of elementary functions. Differential. Numerical differentiation. Parametrically defined functions and their derivatives.
7. Applications of derivatives I: tangent and normal to a curve; Taylor polynomial; extrema of functions; monotonicity.
8. Applications of derivatives II: curvature of a function; inflection points; l'Hôpital's rule; asymptotes; finding inverse functions.
9. What is an integral? Definition of the definite integral as area and the indefinite integral with the concept of an antiderivative. The Riemann integral and the Newton–Leibniz formula. Numerical integration.
10. Integration methods: integration of elementary functions; basic substitution; integration by parts; integration using partial fractions (with real roots); introduction to integration of trigonometric functions.
11. Geometric and engineering applications of the definite integral. Improper integrals.
2. Matrix operations; rank of a matrix; the Frobenius theorem. Computation of the inverse matrix using the Gaussian method. Matrix equations.
3. Determinants; properties of determinants; computation of determinants of degree 2 and 3; Cramer's rule.
4. Real-valued functions of a single real variable: definition, graphs. Properties of real functions; operations with functions; composition of functions; inverse functions.
5. Introduction to limits. Sequences and their limits. Definition of the limit of a function. Continuous and discontinuous functions and their properties. Theorems on continuity.
6. Derivative of a function; its geometric and physical meaning. Differentiation rules for power functions. Linearity of the derivative; derivatives of products, quotients, and composite functions. Derivatives of elementary functions. Differential. Numerical differentiation. Parametrically defined functions and their derivatives.
7. Applications of derivatives I: tangent and normal to a curve; Taylor polynomial; extrema of functions; monotonicity.
8. Applications of derivatives II: curvature of a function; inflection points; l'Hôpital's rule; asymptotes; finding inverse functions.
9. What is an integral? Definition of the definite integral as area and the indefinite integral with the concept of an antiderivative. The Riemann integral and the Newton–Leibniz formula. Numerical integration.
10. Integration methods: integration of elementary functions; basic substitution; integration by parts; integration using partial fractions (with real roots); introduction to integration of trigonometric functions.
11. Geometric and engineering applications of the definite integral. Improper integrals.
E-learning
Literature
[1] BIRD, J. O. Bird's higher engineering mathematics. 9th ed., London: Routledge, Taylor & Francis Group, 2021. ISBN 978-0-367-64375-1.
[2] STRANG, Gilbert. Calculus. Third edition. Wellesley, MA: Wellesley-Cambridge Press, 2017. ISBN 978-0-9802327-5-2 .
[2] STRANG, Gilbert. Calculus. Third edition. Wellesley, MA: Wellesley-Cambridge Press, 2017. ISBN 978-0-9802327-5-2 .
Advised literature
[1] STEWART, J. Calculus: metric version. 8th ed. [Boston]: Cengage Learning, [2016]. ISBN 978-1-305-26672-8.
[2] CROFT, T. et al., Engineering mathematics. Fifth edition. Harlow: Pearson, [2017]. ISBN 978-1-292-14665-2.
[2] CROFT, T. et al., Engineering mathematics. Fifth edition. Harlow: Pearson, [2017]. ISBN 978-1-292-14665-2.