# course: Mathematics 1

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

## dates in winter term

• start: Monday the 02.11.2020
• lecture Tuesdays: from 10:15 to 12.00 o'clock in Online
• lecture Wednesdays: from 10:15 to 12.00 o'clock in Online
• lecture Fridays: from 10:15 to 12.00 o'clock in Online
• tutorial Tuesdays: from 14:15 to 15.45 o'clock in Online
• extra tutorial Mondays: from 11:00 to 12.30 o'clock in Online

## Exams

##### Die Angaben zu den Prüfungsmodalitäten (im WiSe 2020/2021 | SoSe 2021) erfolgen vorbehaltlich der aktuellen Situation. Notwendige Änderungen aufgrund universitärer Vorgaben werden zeitnah bekanntgegeben.
 Form of exam: written Registration for exam: FlexNow Date: 09.04.2021 Begin: 10:00 Duration: 120min description of exam: Prüfung findet online statt.
##### Die Angaben zu den Prüfungsmodalitäten (im WiSe 2020/2021 | SoSe 2021) erfolgen vorbehaltlich der aktuellen Situation. Notwendige Änderungen aufgrund universitärer Vorgaben werden zeitnah bekanntgegeben.
 Form of exam: written Registration for exam: FlexNow Date: 23.08.2021 Begin: 14:30 Duration: 120min Rooms : HGD 30 ,  HMA 10,  HNC 10 Individual appointments of students to each exam location will be issued by the responsible chair.

## 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

1. 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
2. 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
3. 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
4. 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
5. 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
6. 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
7. Ordinary differential equations
• Elementary solving methods of first order ode
• Linear ode (constant coefficients, second order)
8. Orthonormal systems
• Best L^2 approximation, Bessel inequality, Parseval equality
• Real and complex Fourier series
• Complex vector spaces, unitary matrices

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

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