course: Mathematics 2

number:
150112
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:
summer term

dates in summer term

  • start: Monday the 20.04.2020
  • lecture Mondays: from 12:00 to 14.00 o'clock
  • lecture Tuesdays: from 10:00 to 12.00 o'clock
  • lecture Fridays: from 08:00 to 10.00 o'clock
  • tutorial Wednesdays: from 12:00 to 14.00 o'clock
  • extra tutorial Thursdays: from 10:00 to 12.00 o'clock

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:18.08.2020
Begin:10:30
Duration:120min
Rooms : HGA 10,  HGB 10,  HGD 10 ,  HGD 30 ,  HMA 10
Individual appointments of students to each exam location will be issued by the responsible chair.

goals

Proficiency in applying the following mathematical methods:

  • Differential calculus with functions of several real variables
  • Integral calculus with functions of several real variables
  • Definiton and properties of Laplace transform and Fourier transform
  • Complex analysis in one variable

content

  1. Differential calculus with functions of several real variables
    • Functions of several real variables
      • Graph of a function, level sets, continuous functions
    • Differential calculus
      • directional derivative, partial derivative, gradient, total derivative, derivation rules, mean value theorem, higher order derivatives
    • Applications
      • Leibniz integral rule, Taylor series, implicit and inverse functions, Extrema with and without constraints
  2. Integral calculus
    • Riemannian integral
      • definition, mesurable sets, mean value theorem, iterated integrals
    • normal domains, Fubini's theorem, integration by substitution Trägheitsmoment
    • Cavalieri's principle, solid of revolution, center of mass, moment of inertia
    • Improper integrals
      • integrability, sequence of exhaustion
  3. Vector calculus
    • curves
      • definition, parametrisation, tangent vector, length, line integral
    • differential operators rot (curl) and div, conservative vector field, Poincare lemma, vector potential
    • surfaces
      • definition, parametrisation, tangent vector and normal vector, area, surface integral, flux
    • Stokes' theorems
      • Green, Stokes, Gauss
  4. Complex analysis
    • Continuous and holomorphic functions of one complex variable
    • conformal mappings, Möbius transformation
    • Complex curve integral, Cauchy's integral theorem, Cauchy's integral formula, complex antiderivative
    • power series, Laurent series, isolated singularities
    • Residue theorem, application to real integrals
  5. Laplace transform and Fourier transform
    • Laplace transform
      • definition, properties, inverse Laplace transform, application to integral equations
    • Fourier transform
      • definition, properties, inverse Fourier transform

requirements

none

recommended knowledge

Content of lecture "Mathematik 1"

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

Im Sommersemester 2020 wird dieser Kurs bis auf weiteres als online-gestützte Veranstaltung ohne Präsenzveranstaltungen durchgeführt. Die Koordination der Kursaktivitäten wird über Moodle erfolgen.