Title page
Contents (LaTeX)
Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany
Contents (HTML)
Contents (HTML)
Contents (LaTeX)
LEBESGUE Integral
Additive measures
Elementary LEBESGUE measure on
R
Counting measure
Elementary LEBESGUE measure on
R
n
Product measure
Step functions
Properties of the elementary integral
Motivation
Definition (sigma-additivity)
Outer measure and null sets
Properties μ -almost everywhere
Equivalence relation
LEBESGUE measure on
R
n
Lemma (Part 1 of proof)
Theorem (Part 2 of proof)
Proof of theorem without completeness (Part 3)
Theorem (Part 4 of proof)
Completeness (Part 5 of proof)
Measurable sets
Convergence and convergence in measure
Convergence and μ -almost everywhere convergence
Properties of the modulus of a function
Properties of the LEBESGUE measure on
R
n
Proof of the theorem
Extension of measures
Proof
Uniqueness of extension
Measurable functions
Proofs
Proof
Examples
Lemma of FATOU
Dominated convergence theorem
Continuity theorem
Differentiability theorem
FOURIER transformation
Convolution operator
Examples of DIRAC sequences
Approximation by convolution
Approximation theorem of WEIERSTRASS
Gravitational potential
Multiple integrals
Examples as motivation
Proof of the theorem of FUBINI
Examples and the principle of CAVALIERI
Connection between measure and integral
Rotation invariant bodies
Convolution
Approximation by convolution
Transformation of null sets
Proof
Proof of the transformation rule
Special transformations I
Special transformations II
Surfaces in
R
n
Introduction
Simple examples
Representation of tangent spaces
Proof of equivalences
Conclusion
Measure on affine subspaces
Proof of these properties
Computation of surface integrals
Integrals on curves
Hypersurfaces as graph
Rotational symmetric functions
The surface element
Axially symmetric surfaces
Some surface areas
Potential of a surface density
Partial integration in
R
n
Motivation
Interior case
Definition ( C
1
-boundary)
Local case at boundary
Partition of unity
m -dimensional zero sets
Proof of the divergence theorem
Partial Integration
GREEN's formula
CAUCHY's theorem
Isolated singularities
CAUCHY's formula
Representation theorem for LAPLACE operator
Divergence type equations
Functional
Minima of functionals
Tangent cone and directional derivative
First variation
EULER-LAGRANGE equation
Properties
Properties
Properties
Proof
Proof
Proof
Proof
EULER-System für ein Gas
Partial integration on surfaces
Differentiability on surfaces
Remark
Representation of first order operators
Example
Beweis des GAUSS'schen Satzes auf Flächen
Beweis der Eigenschaften der Krümmung
Kurven und Graphen
Beweis des Satzes von STOKES
Beweis zur Variation des Flächeninhalts
Partielle Integration auf Flächen
Darstellung des LAPLACE-BETRAMI-Operators
Appendix: Oriented integrals
Alternating multilinear forms
Wedge product (Outer product) in
R
n
Orientation of subspaces
Differential forms
Hinweis zum Vorlesungsablauf
Exterior derivative
Integration of differential forms
CAUCHY's theorem
General STOKES theorem
GAUSS theorem for differential forms
Classical STOKES theorem for differential forms
GAUSS theorem for oriented surfaces and tangential fields
Direct proof of the general STOKES theorem
Final remark
Exercises
Exercise 1 (Variationsmaß)
Exercise 2 (Treppenfunktionen)
Exercise 3 (Nullmengen)
Exercise 4 (STIELTJES-Maß)
Exercise 5 (Nullmengen)
Exercise 6 (Äußeres Maß)
Exercise 7 ( ϭ -Additivität)
Exercise 8 (Treppenfunktionen)
Exercise 7a (Äußeres Maß)
Exercise 9 (LEBESGUE-Integral für das Zählmaß)
Exercise 10 (Konvergenz in L(μ;Y) )
Exercise 11 (Konvergenz in L(μ;Y) und punktweise Konvergenz)
Exercise 12 (Uneigentliche RIEMANN-Integrale und LEBESGUE-Integral)
Exercise 13 (Verhalten des Integrals bei Streckungen)
Exercise 14 (Messbare Funktionen)
Exercise 15 (Integrierbarkeit von |x|
α
)
Exercise 16 (Unendliche Reihen integrierbarer Funktionen)
Exercise 17 (Oszillierende Funktionen)
Exercise 18 (Zum LEBESGUE'schen Konvergenzsatz)
Exercise 19 (Ein weiteres Konvergenzkriterium)
Exercise 20 (Konvergenz von Integralkernen)
Exercise 21 (FOURIER-Transformation)
Exercise 22 (Dirac-Folgen)
Exercise 23 (Gammafunktion)
Exercise 24 (NEWTON-Potential)
Exercise 25 (Zur Faltung)
Exercise 26 (Masse, Schwerpunkt und Trägheitstensor für ein Simplex)
Exercise 27 (Anwendung von FUBINI)
Exercise 28 (Anwendung von FUBINI)
Exercise 28a (Berechnung dreidimensionaler Integrale)
Exercise 29 (Abschneidefunktion)
Exercise 30 (Fundamentallösung der Wärmeleitungsgleichung)
Exercise 31 (Wärmeleitungsgleichung)
Exercise 32 (Transformationsformel)
Exercise 33 (Zur FOURIER-Transformation)
Exercise 34 (Transformationssatz für eine konforme Abbildung)
Exercise 35 (Spiegelung am Einheitskreis)
Exercise 36 (Zum Transformationssatz)
Exercise 36a (Kardioide)
Exercise 36b (Transformation auf Halbraum)
Exercise 37 (Wiederholungsaufgabe)
Exercise 38 (Wiederholungsaufgabe)
Exercise 39 (Verschiedene Darstellungen eines Hyperboloids)
Exercise 40 (Tangentialkegel)
Exercise 41 (Krümmung einer Kurve)
Exercise 42 (Möbiusband)
Exercise 43 (Catenoid)
Exercise 44 (Zykloid)
Exercise 45 (Äußere Normalen an Kugel und Kegel)
Exercise 46 (Anwendungen zum Satz von GAUSS)
Exercise 47 (Kugelabschnitt)
Exercise 48 (Satz von GAUSS auf unbeschränkten Gebieten)
Exercise 49 (Torsion und FRENET'sches Dreibein)
Exercise 50 (GAUSS'sches Gesetz der Elektrostatik)
Exercise 51 (Zur CAUCHY-Integralformel)
Exercise 52 (Harmonische und holomorphe Funktionen)
Exercise 53 (Transformationsformel auf Flächen)
Exercise 54 (Anwendung der Transformationsformel auf Flächen)
Exercise 55 (Energieerhaltung für die Wellengleichung)
Exercise 56 (Brachistochrone)
Solutions
Tables
Contents (LaTeX)
Greek Alphabet
Partition into lectures
Index
Footnotes
Navigation
Version 1.7
H.W. Alt - 02.01.2007
Title page
Contents (LaTeX)
Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany
