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