Postmodern analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English German |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2005
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XV, 371 S. graph. Darst. 24 cm |
| ISBN: | 9783540258308 3540258302 |
Internformat
MARC
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| 245 | 1 | 0 | |a Postmodern analysis |c Jürgen Jost |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2005 | |
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Datensatz im Suchindex
| _version_ | 1804133582298415104 |
|---|---|
| adam_text | Contents
Chapter I. Calculus for Functions of One Variable
0.
Properties of the real numbers, limits and convergence of sequences of
real numbers, exponential function and logarithm. Exercises
1.
Definitions of continuity, uniform continuity, properties of continuous
functions, intermediate value theorem, Holder and Lipschitz continuity.
Exercises 1
2.
Definitions of differentiability, differentiation rules, differentiable
functions are continuous, higher order derivatives. Exercises
3.
Differential Equations
Characterization of local
mean value theorems, the differential equation
solutions of differential equations, qualitative behavior of solutions of
differential equations and inequalities, characterization of local maxima
and minima via second derivatives, Taylor expansion. Exercises
4.
Banach Space
Banach fixed point theorem, definition of norm, metric, Cauchy
sequence, completeness. Exercises
XII
5. Uniform
Processes. Examples of Banach Spaces. The Theorem
of Arzela-
Convergence of sequences of functions, power series, convergence
theorems, uniformly convergent sequences, norms on function spaces,
theorem of Arzela-Ascoli on the uniform convergence of sequences of
uniformly bounded and equicontinuous functions. Exercises
6.
Primitives, Riemann integral, integration rules, integration by parts,
chain rule, mean value theorem, integral and area, ODEs, theorem of
Picard-Lindelöf
ODEs with a Lipschitz condition. Exercises
Chapter II. Topological Concepts
7.
Compact Sets
Definition of a metric space, open, closed, convex, connected, compact
sets, sequential compactness, continuous mappings between metric spaces,
bounded linear operators, equivalence of norms in TS.d, definition of a
topological space. Exercises r,n
Chapter III. Calculus in Euclidean and Banach Spaces
8.
Definition of differentiability of mappings between Banach spaces,
differentiation rules, higher derivatives, Taylor expansion. Exercises
9.
A. Scalar valued functions
Gradient, partial derivatives, Hessian, local
operator, partial differential equations
B. Vector valued functions
Jacobi matrix, vector fields, divergence, rotation. Exercises
10.
Implicit and inverse function theorems,
Lagrange
Contents
11.
Regular and singular curves, length, rectifiability, arcs,
theorem, higher order ODE as systems of ODEs. Exercises
145
Chapter IV. The Lebesgue Integral
12.
Theorem of Dini, upper and lower semicontinuous functions, the
characteristic function of a set. Exercises n
......................................................... 157
13.
The Volume of Compact Sets
The integral of continuous and semicontinuous functions, theorem of
Pubini,
examples. Exercises
14.
Upper and lower integral, Lebesgue integral, approximation of
Lebesgue integrals, integrability of sets. Exercises 1 „Q
.........................................................
15.
Null functions, null sets, Cantor set, equivalence classes of
functions, the space L1, Fubini s theorem for
Exercises «nr
.........................................................
16.
Theory
Monotone convergence theorem of B.
convergence theorem of H. Lebesgue, parameter dependent integrals,
differentiation under the integral sign. Exercises
17.
The Theorem of Egorov
Measurable functions and their properties, measurable sets,
measurable functions as limits of simple functions, the composition of a
measurable function with a continuous function is measurable, Jensen s
inequality for convex functions, theorem of Egorov on almost uniform
convergence of measurable functions, the abstract concept of a measure.
Exercises
217
XIV
18. The Transformation
Transformation
in polar coordinates. Exercises „oq
Chapter V. IP and Sobolev Spaces
19.
IP -functions, Holder s inequality, Minkowski s inequality,
completeness of ¿p-spaces, convolutions with local kernels, Lebesgue
points, approximation of Lp-functions by smooth functions through
mollification, test functions, covering theorems, partitions of unity.
Exercises
20.
Weak derivatives defined by an integration by parts formula, Sobolev
functions have weak derivatives in Lp-spaces, calculus for Sobolev
functions, Sobolev embedding theorem on the continuity of Sobolev
functions whose weak derivatives are
power,
on the .^-convergence of sequences with bounded Sobolev norm.
Exercises
.........................................................
Chapter VI. Introduction to the Calculus of Variations and
Elliptic Partial Differential Equations
21.
Definition and properties of Hubert spaces, Riesz representation
theorem, weak convergence, weak compactness of bounded sequences,
Banach-Saks lemma on the convergence of convex combinations of
bounded sequences. Exercises
.................................,.......................
22.
Dirichlet s principle, weakly harmonic functions, Dirichlet
problem, Euler-Lagrange equations, variational problems, weak lower
semicontinuity of variational integrals with convex integrands, examples
from physics and continuum mechanics, Hamilton s principle, equilibrium
states, stability, the Laplace operator in polar coordinates. Exercises
23.
Smoothness of weakly harmonic functions and of weak solutions of
general elliptic PDEs, boundary regularity, classical solutions. Exercises
Contents
24. The Maximum
Weak and strong maximum principle for solutions of elliptic PDEs,
boundary point lemma of E.
Liouville. Exercises
25.
Eigenfunctions of the Laplace operator form a complete
basis of L2 as an application of the Rellich compactness theorem.
Exercises
.........................................................
Index
|
| any_adam_object | 1 |
| author | Jost, Jürgen 1956- |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 400 |
| ctrlnum | (OCoLC)61429294 (DE-599)BVBBV020020624 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV020020624 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T20:11:00Z |
| institution | BVB |
| isbn | 9783540258308 3540258302 |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013341998 |
| oclc_num | 61429294 |
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| physical | XV, 371 S. graph. Darst. 24 cm |
| publishDate | 2005 |
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| publisher | Springer |
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| series2 | Universitext |
| spelling | Jost, Jürgen 1956- Verfasser (DE-588)115774564 aut Postmodern analysis Jürgen Jost 3. ed. Berlin [u.a.] Springer 2005 XV, 371 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Universitext Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2668608&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UBRegensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013341998&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jost, Jürgen 1956- Postmodern analysis Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Postmodern analysis |
| title_auth | Postmodern analysis |
| title_exact_search | Postmodern analysis |
| title_full | Postmodern analysis Jürgen Jost |
| title_fullStr | Postmodern analysis Jürgen Jost |
| title_full_unstemmed | Postmodern analysis Jürgen Jost |
| title_short | Postmodern analysis |
| title_sort | postmodern analysis |
| topic | Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | Mathematical analysis Analysis |
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