Calculus and Linear Algebra: Fundamentals and Applications

Preface
Contents
1 Language. Sets
	1.1 Language
	1.2 Sets
	References
2 Numbers and Propositions
	2.1 The Natural Numbers
		2.1.1 Counting Problems
	2.2 Prime Numbers
		2.2.1 Codes and Decoding
	2.3 Integer Numbers
	2.4 Rational Numbers
		2.4.1 Representations of Rational Numbers
		2.4.2 The Numeration
	2.5 The Real Numbers
		2.5.1 Density
		2.5.2 Closure of a Set with Respect to an Operations
	2.6 Abbreviated Notations
		2.6.1 There is at Least One ...
	2.7 The Implication
		2.7.1 Implication and Logical Equivalence
		2.7.2 The Theorem
		2.7.3 Tertium Non Datur
		2.7.4 Proofs in Science
		2.7.5 Visual Proofs
		2.7.6 The Inverse Theorem
		2.7.7 Irrationality of sqrt2
		2.7.8 The Pythagorean School
		2.7.9 Socrates and the Diagonal of the Square
	2.8 The Inductive Method and the Induction Principle
		2.8.1 Necessary Condition. Sufficient Condition
	2.9 Intuition
	2.10 Mathematics and Culture
		2.10.1 On Education
		2.10.2 Individual Study and Work
	References
3 Relations
	3.1 Introduction
	3.2 Cartesian Product of Sets. Relations
	3.3 Binary Relations
		3.3.1 Orderings
		3.3.2 The Power Set
		3.3.3 Total Order
	3.4 Preferences
		3.4.1 Indifference
	3.5 Equivalence Relations
		3.5.1 Partitions of a Set
		3.5.2 Remainder Classes
	References
4 Euclidean Geometry
	4.1 Introduction
	4.2 First Axioms
	4.3 The Axiomatic Method
		4.3.1 Further Axioms of Euclidean Geometry
	4.4 The Refoundation of Geometry
	4.5 Geometric Figures
		4.5.1 Convex and Concave Figures
		4.5.2 Angles
		4.5.3 Relations Between Lines and Planes
		4.5.4 Relations Between Planes
		4.5.5 Projections
		4.5.6 The Angle of a Line and a Plane
		4.5.7 Dihedrals
		4.5.8 Perpendicular Planes
		4.5.9 Symmetries
		4.5.10 Similar Polygons
	4.6 Thales’ Theorem
	References
5 Functions
	5.1 Introduction
	5.2 Equipotent Sets. Infinite Sets, Finite Sets
	5.3 Hotel Hilbert
	5.4 Composite Functions
	5.5 Restriction and Extension of a Function
	Bibliography
6 The Real Line
	6.1 Introduction
	6.2 The Coordinate System of the Axis
		6.2.1 The Measure of a Segment
		6.2.2 The Coordinate System of an Axis
	6.3 Equalities and Identities. Equivalent Equations
		6.3.1 Examples
		6.3.2 Forming an Equation from Given Information
	6.4 Order in R
		6.4.1 Evaluating an Inequality to Making a Decision
	6.5 Intervals, Neighborhoods, Absolute Value
		6.5.1 Exercises
	6.6 The Extended Set of Real Numbers
		6.6.1 Examples
	6.7 Upper Bounds and Lower Bounds
	6.8 Commensurability and Real Numbers
	6.9 Separate Sets and Contiguous Sets
	Bibliography
7 Real-Valued Functions of a Real Variable. The Line
	7.1 The Cartesian Plane
		7.1.1 Quadrants
		7.1.2 Distance
	7.2 Real-Valued Functions of a Real Variable
		7.2.1 Extrema of a Real-Valued Function
		7.2.2 The Graph of a Real-Valued Function
		7.2.3 Graph and Curve
	7.3 Lines in the Cartesian Plane
		7.3.1 The Constant Function
		7.3.2 The Identical Function
		7.3.3 The Function f ∶ x→ kx
		7.3.4 The Function f ∶ x → kx + n
		7.3.5 The Linear Equation
		7.3.6 The Parametric Equations of the Line
	7.4 Parallel Lines
		7.4.1 Parallel Lines Represented by Parametric Equations
		7.4.2 Parallel Lines Represented by Ordinary Equations
		7.4.3 Parallel Lines. Exercises
	7.5 The Absolute Value Function
	7.6 A Linear Model
	7.7 Invertible Functions and Inverse Functions
	Bibliography
8 Circular Functions
	8.1 Introduction
		8.1.1 The Equation of the Circumference
		8.1.2 The Goniometric Circumference
		8.1.3 Sine, Cosine and Tangent
		8.1.4 Further Goniometric Identities
		8.1.5 The Graphs of Sinx, Cosx and Tanx
	Bibliography
9 Geometric and Numeric Vectors
	9.1 n-Tuples of Real Numbers
		9.1.1 Linear Combinations of n-Tuples
	9.2 Scalars and Vectors
	9.3 Applied Vectors and Free Vectors
	9.4 Addition of Free Vectors
	9.5 Multiplication of a Scalar by a Free Vector
	9.6 Properties of Operations with Free Vectors
	9.7 Component Vectors of a Plane Vector
	9.8 Space Coordinate System and Vectors
	9.9 Unit Vectors
	9.10 The Sphere
	Bibliography
10 Scalar Product. Lines and Planes
	10.1 Introduction
	10.2 Scalar Product
		10.2.1 Orthogonal Projections of a Vector
		10.2.2 Scalar Product in Terms of the Components
	10.3 Scalar Product and Orthogonality
		10.3.1 Angles of Lines and Vectors
		10.3.2 Orthogonal Lines in the Plane
	10.4 The Equation of the Plane
	10.5 Perpendicular Lines and Planes
	Bibliography
11 Systems of Linear Equations. Reduction
	11.1 Linear Equations
		11.1.1 Systems of Linear Equations
	11.2 Equivalent Systems
		11.2.1 Elementary Operations
	11.3 Reduced Systems
	11.4 Exercises
	Bibliography
12 Vector Spaces
	12.1 Introduction
		12.1.1 Complex Numbers
	12.2 Operations
	12.3 Fields-
	12.4 Vector Spaces
	12.5 Linear Dependence and Linear Independence
	12.6 Finitely Generated Vector Spaces. Bases
	12.7 Vector Subspaces
		12.7.1 Spanned Subspaces
	12.8 Dimension
	12.9 Isomorphism
	12.10 Identification of Geometric and Numerical Vector Spaces
	12.11 Scalar Product in Rn
	12.12 Exercises
	Bibliography
13 Matrices
	13.1 First Concepts
	13.2 Reduced Matrices
	13.3 Rank
	13.4 Matrix Reduction Method and Rank
	13.5 Rouché-Capelli’s Theorem
	13.6 Compatibility of a Reduced System
	13.7 Square Matrices
	13.8 Exercises
	Bibliography
14 Determinants and Systems of Linear Equations
	14.1 Determinants
	14.2 Properties of the Determinants
	14.3 Submatrices and Minors
	14.4 Cofactors
	14.5 Matrix Multiplication
	14.6 Inverse and Transpose Matrices
	14.7 Systems of Linear Equations and Matrices
	14.8 Rank of a Matrix and Minors
		14.8.1 Matrix A Has a Non-zero Minor of Maximal Order
		14.8.2 Calculating the Rank of a Matrix Via Kronecker’s Theorem
	14.9 Cramer’s Rule
		14.9.1 Homogeneous Linear Systems
		14.9.2 Associated Homogeneous Linear System
	14.10 Exercises
	Bibliography
15 Lines and Planes
	15.1 Introduction
	15.2 Parallel Lines
	15.3 Coplanar Lines and Skew Lines
	15.4 Line Parallel to Plane. Perpendicular Lines. Perpendicular Planes
	15.5 Intersection of Planes
	15.6 Bundle of Planes
	Bibliography
16 Algorithms
	16.1 Introduction
	16.2 Greatest Common Divisor: The Euclidean Algorithm
	16.3 Regular Subdivision of a Segment
	16.4 Gauss Elimination
	16.5 Conclusion
	Bibliography
17 Elementary Functions
	17.1 Introduction
	17.2 Monotonic Functions
	17.3 Invertible Functions and Inverse Functions
	17.4 The Power
		17.4.1 Power with Natural Exponent
		17.4.2 Power with Non-Zero Integer Exponent
		17.4.3 Null Exponent
	17.5 Even Functions, Odd Functions
	17.6 The Root
		17.6.1 Further Properties of the Power with Rational Exponent
	17.7 Power with Real Exponent
	17.8 The Exponential Function
		17.8.1 Properties of the Exponential Function
		17.8.2 The Number of Napier
	17.9 The Logarithm
	17.10 Conclusion
	17.11 Exercises
	Bibliography
18 Limits
	18.1 Introduction
	18.2 Definition
		18.2.1 Specific Applications of Definition 18.1
		18.2.2 Uniqueness of the Limit
	18.3 Limits of Elementary Functions
	18.4 Properties of Limits
		18.4.1 Operations
		18.4.2 Permanence of the Sign
		18.4.3 Comparison
		18.4.4 Limit of the Composite Function
		18.4.5 Right and Left Limits
		18.4.6 More on the Limits of Elementary Functions
		18.4.7 Solved Problems
		18.4.8 Supplementary Problems
	18.5 Asymptotes
		18.5.1 Vertical Asymptotes
		18.5.2 Horizontal Asymptotes
		18.5.3 Oblique Asymptotes
	Bibliography
19 Continuity
	19.1 Continuous Functions
	19.2 Properties of Continuous Functions
		19.2.1 Uniform Continuity
	19.3 Discontinuity
	19.4 Domain Convention
	19.5 Curves
	19.6 Continuous Functions and Inverse Functions
	19.7 The Inverse Functions of the Circular Functions
	19.8 Continuity of Elementary Functions
	19.9 Solved Problems
	Bibliography
20 Derivative and Differential
	20.1 Introduction
	20.2 Definition of Derivative
	20.3 Geometric Meaning of the Derivative
	20.4 First Properties
		20.4.1 Derivatives of Some Elementary Functions
	20.5 Operations Involving Derivatives
	20.6 Composite Functions. The Chain Rule
		20.6.1 Derivatives of Some Elementary Functions
	20.7 Derivatives of the Inverse Functions
		20.7.1 Derivatives of the Inverses of the Circular Functions
	20.8 The Derivative of the Function (f(x))g(x) and the Power Rule
		20.8.1 Summary of Formulas and Differentiation Rules
	20.9 Right and Left-Hand Derivatives
	20.10 Higher Order Derivatives
	20.11 Infinitesimals
	20.12 Infinities
	20.13 Differential
		20.13.1 Differentials of Higher Order
	20.14 Solved Problems
	Bibliography
21 Theorems of Differential Calculus
	21.1 Introduction
	21.2 Extrema of a Real-Valued Function of a Single Variable
	21.3 Fermat’s and Rolle’s Theorems
	21.4 Lagrange’s Theorem and Consequences
	21.5 Comments on Fermat’s Theorem
		21.5.1 Searching for Relative Maximum and Minimum Points
		21.5.2 Searching for the Absolute Maximum and Minimum of a Function
	21.6 de l’Hospital’s Rule
	21.7 More on the Indeterminate Forms
	21.8 Parabola with Vertical Axis
	21.9 Approximation
		21.9.1 Quadratic Approximation
	21.10 Taylor’s Formula
	21.11 Convexity, Concavity, Points of Inflection
		21.11.1 Convexity and Concavity
		21.11.2 Points of Inflection
		21.11.3 Defiladed Objects
	21.12 Drawing the Graph of a Function
	21.13 Solved Problems
	Bibliography
22 Integration
	22.1 Introduction
	22.2 The Definite Integral
	22.3 Area of a Plane Set
	22.4 The Definite Integral and the Areas
	22.5 The Integral Function
	22.6 Primitive Functions
	22.7 The Indefinite Integral
		22.7.1 Indefinite Integral Calculation
		22.7.2 Some Immediate Indefinite Integrals
		22.7.3 A Generalization of Indefinite Integration Formulas
	22.8 Integration by Parts
		22.8.1 Indefinite Integration Rule by Parts
		22.8.2 Definite Integration Rule by Parts
	22.9 Area of a Normal Domain
	22.10 Trigonometric Integrals
		22.10.1 Trigonometric Substitutions
	22.11 Improper Integrals
		22.11.1 Improper Integrals Over Bounded Intervals
		22.11.2 Improper Integrals Over Unbounded Intervals
	22.12 Problems Solved. Indefinite and Improper Integrals
	Bibliography
23 Functions of Several Variables
	23.1 Introduction
	23.2 The Real n-Dimensional Space
	23.3 Examples of Functions of Several Variables
	23.4 Real-Valued Functions of Two Real Variables
	23.5 More About the Domain of f(x, y)
	23.6 Planes and Surfaces
	23.7 Level Curves
	23.8 Upper Bounded and Lower Bounded Functions
	23.9 Limits
	23.10 Continuity
	23.11 Partial Derivatives
	23.12 Domains and Level Curves
	23.13 Solved Problems
	23.14 Partial Derivatives of the Functions of Several Variables
	23.15 Partial Derivatives of Higher Order
	Bibliography
24 Curves and Implicit Functions
	24.1 Curves and Graphs
	24.2 Regular Curves
		24.2.1 Tangent Line to a Regular Curve
	24.3 Closed Curves
	24.4 Length of a Curve
		24.4.1 Problems
	24.5 Curvilinear Abscissa
	24.6 Derivative of the Composite Functions
	24.7 Implicit Functions
		24.7.1 Higher Order Derivatives
	Bibliography
25 Surfaces
	25.1 Introduction
	25.2 Cylinder
	25.3 Cone
		25.3.1 Homogeneous Polynomial
	25.4 Exercises
	Bibliography
26 Total Differential and Tangent Plane
	26.1 Introduction
	26.2 Total Differential
	26.3 Vertical Sections of a Surface
	26.4 The Tangent Plane to a Surface
	Bibliography
27 Maxima and Minima. Method of Lagrange Multipliers
	27.1 Relative and Absolute Extrema of Functions of Two Variables
	27.2 Exercises
	27.3 Search for Relative Maxima and Minima
	27.4 Absolute Maxima and Minima in R2
	27.5 Search for Extrema of a Continuous Function
	27.6 Constrained Extrema. Method of Lagrange Multiplier
	27.7 Method of Lagrange Multipliers
	Bibliography
28 Directional Derivatives and Gradient
	28.1 Directional Derivatives
	28.2 Gradient
		28.2.1 Steepest Descent
	Bibliography
29 Double Integral
	29.1 Area of a Plane Set
	29.2 Volume of a Solid
	29.3 Cylindroid
	29.4 Double Integral
	29.5 Properties of the Double Integral
	29.6 Double Integral Reduction Formulas
	Bibliography
30 Differential Equations
	30.1 Introduction
	30.2 Separable Equations
	30.3 Exponential Growth and Decay
	Bibliography
Index