# course: Mathematics 2

number:
150112
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:
summer term

## dates in summer term

• start: Monday the 12.04.2021
• 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 10:00 to 12.00 o'clock
• tutorial Wednesdays: from 12:00 to 14.00 o'clock
• tutorial Thursdays: from 08:00 to 10.00 o'clock
• extra tutorial Thursdays: from 10:00 to 12.00 o'clock

## 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: 01.06.2021 Begin: 16:30 Duration: 120min description of exam: verschobener Prüfungstermin WiSe 20/21 Rooms : HZO 30,  HZO 40,  HZO 60 Individual appointments of students to each exam location will be issued by the responsible chair.
##### 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: 14.09.2021 Begin: 16:30 Duration: 120min Rooms : Mensa Mitte ,  Mensa Nord Ost ,  Mensa Nord West ,  Mensa West ,  Mensa Ost 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

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 2021 wird dieser Kurs bis auf weiteres als online-gestützte Veranstaltung ohne Präsenzveranstaltungen durchgeführt. Die Koordination der Kursaktivitäten wird über Moodle erfolgen.