Elementary Analysis

Elementary Analysis

The Commonwealth and International Library: Mathematics Division, Volume 2

1st Edition - January 1, 1966

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  • Authors: K. S. Snell, J. B. Morgan
  • eBook ISBN: 9781483158983

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Description

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.

Table of Contents


  • 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, cos2θ+ in2θ=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


Product details

  • No. of pages: 304
  • Language: English
  • Copyright: © Pergamon 1966
  • Published: January 1, 1966
  • Imprint: Pergamon
  • eBook ISBN: 9781483158983

About the Authors

K. S. Snell

J. B. Morgan

About the Editors

W. J. Langford

E. A. Maxwell

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