This book gives many helps for students of technical colleges who have had usual mathematical training. The material presented in this book exceeds the content of the spoken lessons, and so, it is also useful for other engineering specialities and even for students in mathematics.
The authors present in a small number of pages the basic notions and results of differential calculus concerning to: sequences and series of numbers, sequences and series of functions, power series, elements of topology in n-dimensional space, limits of functions, continuous functions, partial derivatives of functions of several variables, Taylor's formula, extrema of a function of several variables (free or with constrains), change of variables, dependent functions.
Author(s): Gavriil Paltineanu, Ileana Bucur, Mariana Zamfir
Publisher: Springer
Year: 2022
Language: English
Pages: 190
City: Singapore
Preface
Contents
1 Sequences of Real Numbers
1.1 Real Numbers
1.2 Real Number Sequences
1.3 Extended Real Number Line
2 Real Number Series
2.1 Convergent and Divergent Series
2.2 Series with Positive Terms
2.3 Series with Arbitrary Terms
2.4 Approximating the Sum of a Leibniz’s Series
2.5 Absolutely and Conditionally Convergent Series
2.6 Operations on Convergent Series
2.7 Sequences and Series of Complex Numbers
3 Sequences of Functions (Functional Sequences)
3.1 Simple and Uniformly Convergence
3.2 The Properties of the Uniformly Convergent Functional Sequences
4 Series of Functions (Functional Series)
4.1 Simple and Uniform Convergence
4.2 Properties of the Uniformly Convergent Series of Functions
4.3 Power Series
4.4 Taylor’s Formula
4.5 Taylor’s and Maclaurin’s Series
4.6 Elementary Functions. Euler’s Formulas. Hyperbolic Trigonometric Functions
5 Functions of Several Variables
5.1 Vector Space mathbbRn. Basic Notions and Notations
5.2 Convergent Sequences of Vectors in mathbbRn
5.3 Topology Elements on mathbbRn
5.4 Limits of Functions of Several Variables
5.5 Continuous Functions of Several Variables
5.6 Properties of Continuous Functions Defined on Compact or Connected Sets
5.7 Linear Continuous Maps from mathbbRn to mathbbRm
6 Differential Calculus of Functions of Several Variables
6.1 Partial Derivatives. Differentiability of a Function of Several Variables
6.2 Differentiability of Vector Functions. Jacobian Matrix
6.3 Differentiability of Composite Functions
6.4 The First Order Differenential and Its Invariance Form
6.5 The Directional Derivative. The Differential Operators: Gradient, Divergence, Curl and Laplacian
6.6 Partial Derivatives and Differentials of Higher Orders
6.7 Second-Order Partial Derivatives of Functions Composed of Two Variables
6.8 Change of Variables
6.9 Taylor’s Formula for Functions of Several Variables
6.10 Local Extrema of a Function of Several Variables
6.11 Local Inversion Theorem
6.12 Regular Transformations
6.13 Implicit Functions
6.14 Local Conditional Extremum
6.15 Dependent and Independent Functions
References
Index
