Table of Contents
Table of Contents
“Introduction to Advanced Mathematical Analysis”
INTRODUCTION
REAL FUNCTIONS WITH MULTIPLE VARIABLES
IMPLICIT FUNCTIONS
INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES
DEVELOPMENTS IN SERIES
COMPLEX ANALYSIS
“Introduction to Advanced Mathematical Analysis”
“Introduction to Advanced Mathematical Analysis”
SIMON MALACRIDA
The following mathematical topics are presented in this book:
real functions with several variables
implicit functions
integral calculus for functions of several variables
developments in power series, Taylor series and Fourier series
analysis in the complex field
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – REAL FUNCTIONS WITH MULTIPLE VARIABLES
Introduction
Operations
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II – IMPLICIT FUNCTIONS
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III - INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES
Introduction
Surface and volume integrals
Remarkable theorems
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IV - DEVELOPMENTS IN SERIES
Convergence criteria for numerical series
Sequence and series of functions
Power series
Taylor and Maclaurin series
Fourier series
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V - COMPLEX ANALYSIS
Property
Monodromy and polydromy
Complex integration
Euler functions
Complex series
INTRODUCTION
INTRODUCTION
The exposition of mathematical analysis does not stop at the introduction of the concepts of neighbourhood, limit, derivative, integral and at the study of real functions with one variable.
These first notions are only a precondition for other much more advanced concepts and, as such, subsequent not only on a cognitive level but also on an applied level.
Real functions with several variables and implicit functions are a first possible extension, as is integral calculus with several variables.
The two fundamental points, however, are given by the developments in series and by the complex analysis.
The series expansion of a function can be done in many ways and this leads to different mathematical and scientific applications.
Power series, Taylor series, and Fourier series are very powerful and effective symbolisms.
On the other hand, complex analysis makes it possible to extend everything studied in the set of real numbers to that of complex numbers, with considerable benefits in terms of general results.
What is stated in this manual is essential for understanding and solving differential equations and functional analysis problems.
For this reason, the topics presented are generally addressed in advanced mathematical analysis courses (2 and 3).
I
REAL FUNCTIONS WITH MULTIPLE VARIABLES
REAL FUNCTIONS WITH MULTIPLE VARIABLES
Introduction
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Functions of real variables with several variables are an extension of what has been said for real functions with one variable.
Almost all the properties mentioned for one-variable functions remain valid (such as injectivity, surjectivity and bijectivity), except the ordering property which is not definable.
The domain of a multivariate function is given by the Cartesian product of the domains calculated on the single variables.
A level set , or level curve, is the set of points such that:
The level set with c=0 is used to analyze the sign of the function in the domain.
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Operations
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The topological definition of limit is the same as that given for one-variable functions, the metric definition changes as follows:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3945-5
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