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

  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

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

  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

miscellaneous