Analysis Course vol. 2
Authors
Description
In this book, functions of n real variables are studied. It begins with the Topology of Euclidean Space, continues with Differential and Integral Calculus in n dimensions and concludes with differential forms on surfaces of Rn. It deals with the topics that make up the old Vector Analysis, which revolve around the Stokes Theorem in its various forms, ending with an exposition of topological degree theory. Its numerous exercises are applications and extensions of the subjects presented, as well as connections with nearby theories.
Name: Analysis Course vol. 2
Author(s): e Elon Lages Lima
Pages: 464
Publication: IMPA, 2020
ISBN: 978-85-244-0494-8
Edition: 12
1 Topology of Euclidean Space
1.1 The vector space R n
1.2 Inner product and norm
1.3 Complex numbers
1.4 Balls and limited sets
1.5 Sequences in Euclidean space
1.6 Accumulation points
1.7 Continuous applications
1.8 Homeomorphisms
1.9 Limits
1.10 Open sets
1.11 Closed sets
1.12 Compact sets
1.13 Distance between two assemblies; diameter
1.14 Connectivity
1.15 The norm of a linear transformation
2 Paths in Euclidean space
2.1 Differentiable paths
2.2 Integral of a path
2.3 The classical theorems of calculus
2.4 Rectifiable paths
2.5 Arc length as a parameter
2.6 Curvature and torsion
2.7 The angle function
3 Real Functions of n Variables
3.1 Partial derivatives
3.2 Directional derivatives
3.3 Differentiable functions
3.4 The differential of a function
3.5 The gradient of a differentiable function
3.6 Leibniz’s rule
3.7 Schwarz’s theorem
3.8 Taylor’s formula: critical points
3.9 The implicit function theorem
3.10 Lagrange’s multiplier
4 Curvilinear integrals
4.1 Differential forms of degree 1
4.2 Stieltjes integral
4.3 Integral of a shape along a path
4.4 Juxtaposition of paths: inverse path
4.5 Curvilinear integral of a vector field and a function
4.6 Exact forms and closed forms
4.7 Homotopy
4.8 Curvilinear integrals and homotopy
4.9 Cohomology
4.10 Kronecker’s formula
5 Differentiable applications
5.1 Differentiability of an application
5.2 Examples of differentiable applications
5.3 The chain rule
5.4 Taylor’s formula
5.5 The inequality of the mean value
5.6 Sequences of differentiable applications
5.7 Strongly differentiable applications
5.8 The inverse application theorem
5.9 Application: Morse’s lemma
5.10 The local form of immersions
5.11 The local form of submersions
5.12 The rank theorem
5.13 Surfaces in Euclidean space
5.14 Steerable surfaces
5.15 The Lagrange multiplier method
6 Multiple integrals
6.1 The definition of an integral
6.2 Sets of zero measure
6.3 Characterization of integrable functions
6.4 The integral as the limit of Riemann sums
6.5 Repeated integration
6.6 Change of variables
7 Surface integrals
7.1 Alternating forms
7.2 Differential forms
7.3 The exterior differential
7.4 Partitions of the unit
7.5 Applications of the partition of the unit
7.6 Surface integrals
7.7 Surfaces with an edge
7.8 Stokes’ theorem
7.9 Degree of an application
7.10 Kronecker’s integral
Bibliography
Index
