Elementary Analysis, Volume 2 introduces several of the ideas of modern mathematics in a casual manner and provides the practical experience in algebraic and analytic operations that lays a sound foundation of basic skills.
This book focuses on the nature of number, algebraic and logical structure, groups, rings, fields, vector spaces, matrices, sequences, limits, functions and inverse functions, complex numbers, and probability. The logical structure of analysis given through the treatment of differentiation and integration, with applications to the trigonometric and logarithmic functions, is also briefly discussed.
This volume begins with a description of the trigonometric functions of the general angle and an introduction to the binomial theorem and series. The rest of the chapters cover the numerical solution of equations, analytical geometry, Argand Diagram, numerical methods, and methods of approximation that form an important section of modern applied mathematics.
This publication is valuable to teachers and students in training colleges.
Inhalt
Preface
Introduction: A summary of Basic Ideas from Volume 1
15. Trigonometric Functions of the General Angle
Angles
Vectors
Trigonometrical Ratios
Special Angles
The General Angle
Periodic Functions
Graphs
General Formula, cos2 + in2 =1
Inequalities
16. Functions of Compound Angles
Unit Vectors
Compound Angles
Multiple Angles
Sums and Products
Even and Odd Functions
Parameters
Inverse Functions
The Circle
17. Differentiation and Integration of the Trigonometric Functions
Gradient of the Sine Function
Radians and Degrees
Differentiation of a Vector
Motion in a Circle
Inequalities
Derivative of Tan x
Integration
Substitution
Polar Coordinates
Area of a Sector
18. Applications of the Trigonometric Functions
Dot Product of Vectors
Compound Angle Formula
Triangle Formula and Problems
Simple Harmonic Motion
Approximations
Three Dimensions
Projections
19. Polynomials
The Remainder Theorem
The Factor Theorem
The Identity Theorem
Synthetic Division
Evaluation of a Polynomial
Repeated Factors
20. Symmetric Functions of the Roots of a Polynomial Equation
Complex Roots
Symmetric Functions
Sums of Powers of Roots
Approximation for the Greatest Root
21. The Binomial Theorem
The Expansion of (a + x)n
The Binomial Theorem
Approximations
Proof by Induction
22. The Binomial Series
Limit Sum
The Binomial Series
Partial Fractions
Expansions
Approximations
23. Numerical Solution of Equations
Linear Interpolation
Newton's Method
Trigonometric Equations
Recurrence Relations and Iterative Methods
Expansions
24. Analytical Geometry
Vectors-A Recapitulation
Coordinates as Vectors
Transformations
Change of Axes
Polar Coordinates
The Parabola, Hyperbola, Circle and Ellipse
Orthogonal Projection
25. The Argand Diagram
Lengths in the Argand Diagram
Geometrical Illustrations
Inversion
Complex Equation of a Line and Circle
Similarity
Conformal Transformations
The Equation xn=1
26. Matrices and Determinants
Successive Transformations
Definition of a Matrix
Addition
Product
Applications
Unit Matrix
Solution of Linear Equations
Elementary Row Operations
The Inverse Matrix
Determinants
Cofactors
Singular Matrices
Quadratic Forms
27. The Exponential and Logarithmic Functions
Limit of a Function
Continuous Functions
Definite Integrals
The Mean Value Theorem for Integrals
The Fundamental Theorem
The Logarithmic Function
The Logarithmic Series
The Exponential Function and Series
Logarithmic Differentiation
Integration of Rational, Algebraic and Trigonometric Functions
Integration by Parts
28. Introduction to Probability Theory
The Concept of Probability
A Sample Space
Alternative or Disjoint Events
Combined Events
Permutations
Combinations
Set Theory and Probability
Independent Events
Tree Diagrams
Binomial Distribution
The Value of an Expectation
Answers
Index