# course: Mathematics 1

- number:
- 150110
- teaching methods:
- lecture with tutorials
- media:
- black board and chalk
- responsible person:
- Dr. rer. nat. Annett Püttmann
- lecturer:
- Dr. rer. nat. Annett Püttmann (Mathematik)
- language:
- german
- HWS:
- 8
- CP:
- 10
- offered in:
- winter term

## dates in winter term

- start: Thursday the 10.10.2019
- lecture Tuesdays: from 10:15 to 12.00 o'clock in HZO 30
- lecture Wednesdays: from 10:15 to 12.00 o'clock in HZO 10
- lecture Fridays: from 10:15 to 12.00 o'clock in HZO 30
- tutorial (alternativ) Thursdays: from 10:15 to 12.00 o'clock in NB 02/99
- tutorial (alternativ) Thursdays: from 10:15 to 12.00 o'clock in NC 3/99
- tutorial (alternativ) Thursdays: from 14:15 to 16.00 o'clock in NB 3/99
- tutorial (alternativ) Thursdays: from 14:15 to 16.00 o'clock in NC 3/99
- tutorial (alternativ) Thursdays: from 16:15 to 18.00 o'clock in NB 2/99
- tutorial (alternativ) Thursdays: from 16:15 to 18.00 o'clock in ND 3/99
- tutorial (alternativ) Thursdays: from 16:15 to 18.00 o'clock in NB 3/99
- tutorial (alternativ) Fridays: from 08:15 to 10.00 o'clock in ND 5/99
- tutorial (alternativ) Fridays: from 08:15 to 10.00 o'clock in NC 3/99
- tutorial (alternativ) Fridays: from 08:15 to 10.00 o'clock in NC 6/99
- extra tutorial Thursdays: from 08:30 to 10.00 o'clock in HID

## Exam

##### All statements pertaining to examination modalities (for the summer/winter term of 2020) are given with reservations. Changes due to new requirements from the university will be announced as soon as possible.

Form of exam: | written |

Registration for exam: | FlexNow |

Date: | 27.07.2020 |

Begin: | 13:30 |

Duration: | 120min |

Room : | HZO 20 |

## goals

Proficiency in applying the following mathematical methods to engineering problems:

- Properties of real and complex numbers
- Elementary concepts of linear algebra
- Differentiation and integration of univariate functions
- Basic ordinary differential equations
- Orthonormal systems, esp. Fourier series

## content

- Real and complex numbers
- Construction of N, Z, Q, R; basic arithmetic rules; order; absolute value (triangle inequality), max, min, sup, inf
- Mathematical description of sets and propositions
- Sum and product symbols; binomial coefficients; binomial theorem; mathematical induction
- Number representation in different bases; binary numbers
- Complex numbers
- Complex plane
- Basic arithmetic rules
- Absolute value, complex conjugation
- Polar coordinates
- Complex powers and roots

- Elementary functions
- Polynomials and rational functions
- Roots, polynomial long division, partial fraction decomposition

- Trigonometric functions (unit circle, angle sum/difference identities)
- Classes of growth
- Composition of functions, transformation/scaling of graphs

- Polynomials and rational functions
- Sequences, continuity, series
- Convergence, limit, algebraic limit theorems, examples
- Definition of continuity, properties of continuous functions, (counter-)examples
- Applications: existence of extrema, intermediate value theorem, existence of roots

- Properties of series, convergence tests

- Differential calculus
- Definition and rules of differentiation, examples (polynomials, rational and trigonometric functions)
- Higher order derivatives, mean value theorem, rule of de l’Hospital, Taylor polynomials, power series, radius of convergence, examples
- Monotonicity, finding extrema, existence and differentiation of the inverse function

- Integral calculus
- Riemann integral, integrability
- Fundamental theorem of calculus
- Antiderivatives, rules of integration, mean value theorem for definite integrals
- Definition and properties of logarithms, Euler number, powers with real exponent
- Integration of sequences and series of functions
- Improper integrals, Gamma function

- Linear algebra
- Real vector space (definition, scalar product, norm, linear independence, dimension)
- Lines, planes, distances, cross product
- Matrices and linear mappings, determinants and inverse linear mappings, change of coordinates, trace
- Systems of linear equations, Gaussian algorithm, calculation of inverse matrices
- Eigenvalues, eigenspaces, diagonalization
- Ellipses, hyperbolas, parabolas

- Ordinary differential equations
- Elementary solving methods of first order ode
- Linear ode (constant coefficients, second order)

- Orthonormal systems
- Best L^2 approximation, Bessel inequality, Parseval equality
- Real and complex Fourier series
- Complex vector spaces, unitary matrices

## requirements

None

## recommended knowledge

Solid foundation in school mathematics. It is highly recommended to attend the pre-course “Mathematik für Ingenieure und Naturwissenschaftler” offered each year in September by the faculty of mathematics.

## literature

- Meyberg, K., Vachenauer, P. "Höhere Mathematik 2", Springer, 2007
- Burg, Klemens, Haf, Herbert, Wille, Friedrich "Höhere Mathematik für Ingenieure 3. Gewöhnliche Differentialgleichungen, Distributionen, Integraltransformationen", Teubner Verlag, 2002
- Meyberg, K., Vachenauer, P. "Höhere Mathematik I", Springer, 1995

## miscellaneous

Hier finden Sie alle wichtigen Informationen: