Sixth Form Pure Mathematics

Sixth Form Pure Mathematics

Volume 2

1st Edition - January 1, 1963

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  • Authors: C. Plumpton, W. A. Tomkys
  • eBook ISBN: 9781483140889

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Description

Sixth Form Pure Mathematics, Volume 2, provides an introduction to inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, and a range of mathematical methods including the use of determinants, the manipulation of inequalities, the solution of easy differential equations, and the use of approximate numerical methods. Complex numbers are defined and the various ways of representing and manipulating them are considered. Polar coordinates, curvature, an elementary study of lengths of curves and areas of surfaces of revolution, a more mature discussion of two-dimensional coordinate geometry than was possible in Volume 1, and an elementary introduction to the methods of three dimensional coordinate geometry comprise the geometrical content of the book. Throughout, the authors have tried to preserve the concentric style which they used in Volume 1 and the many worked examples and exercises in each chapter are designed or chosen to provide a continuous reminder of the work of the preceding chapters. Except for Pure Geometry, the two volumes cover almost all of the syllabuses for Advanced Pure Mathematics of the nine Examining Boards. This book provides an adequate course for mathematical pupils at Grammar Schools and a useful introductory course for Science and Engineering students in their first year at University or Technical College or engaged in private study.

Table of Contents


  • Chapter XI Linear Equations and Determinants

    The Solution of Linear Simultaneous Equations in Two Unknowns

    Second Order Determinants

    Rules of Manipulation for Second Order Determinants

    Third Order Determinants

    Factorization of Determinants

    Geometrical Interpretation.

    The Equation of a Straight Line through Two Given Points

    The Equation of a Tangent to the Curve

    The Equation of a Line-Pair

    The Equation of a Circle through Three Given Points

    The Area of a Triangle

    The Solution of Simultaneous Equations

    Summary

    The Product of Two Determinants

    The Derivative of a Determinant

    Chapter XII Inverse Circular Functions, Hyperbolic Functions and Inverse Hyperbolic Functions

    Inverse Circular Functions.

    The Derivatives of the Inverse Circular Functions

    Standard Integrals

    The Hyperbolic Functions

    Inverse Hyperbolic Functions

    Derivatives of the Inverse Hyperbolic Functions

    Logarithmic Forms of the Inverse Hyperbolic Functions

    Methods of Integration

    The Integrals

    Summary of Standard Integrals and Methods of Integration So Far Considered

    Chapter XIII Definite Integrals. Further Applications of Integration

    Properties of Definite Integrals

    Infinite Integrals

    Reduction Formula

    Approximate Numerical Integration

    Mean Values and Root Mean Square

    Center of Mass

    The Theorem of Pappus concerning Volumes

    Moments of Inertia

    Chapter XIV Some Properties of Curves

    Points of Inflexion

    The Length of a Curve

    The Cycloid

    Areas of Surfaces of Revolution

    The Theorem of Pappus concerning Surfaces of Revolution

    Curvature

    Newton's Formula for Radius of Curvature at the Origin

    Chapter XV Polar Coordinates

    Definitions

    Loci in Polar Coordinates

    Curve Sketching in Polar Coordinates

    The Lengths of Chords of Polar Curves Which are Drawn through The Pole

    Transformations from Polar to Cartesian Equations and the Reverse Process

    Areas in Polar Coordinates

    The Length of an Arc in Polar Coordinates

    Volumes of Revolution and Areas of Surfaces of Revolution in Polar Coordinates

    The Angle between the Tangent and the Radius Vector

    The Tangential Polar Equation—Curvature

    Chapter XVI Complex Numbers

    The Number System

    Definition of Complex Number

    The Cube Roots of Unity

    Conjugate Pairs of Complex Roots

    The Geometry of Complex Numbers

    The Polar Coordinate Form of a Complex Number- Modulus and Argument

    Products and Quotients

    De Moivre's Theorem

    The Exponential Form of a Complex Number

    Exponential Values of Sine and Cosine

    Chapter XVII Differential Equations

    Formation of Differential Equations

    The Solution of a Differential Equation

    First Order Differential Equations with Variables Separable

    Homogeneous Equations

    The Law of Natural Growth

    Linear Equations of the First Order

    Bernoulli's Equation

    Equations of Higher Orders

    Linear Equations of the Second Order with Constant Coefficients

    The Complementary Function. The Particular Integral

    Chapter XVIII Approximate Numerical Solution of Equations

    Graphical Methods

    The Number of Real Roots of An Equation

    The Approximate Value of a Small Root of a Polynomial Equation

    Newton's Method for Obtaining a Closer Approximation to a Real Root of An Equation

    Chapter XIX Inequalities

    Rules of Manipulation

    Fundamental Inequalities

    The Calculus Applied to Inequalities

    Chapter XX Coordinate Geometry

    The Straight Line

    Lone Pairs

    The Circle

    The Radical Axis

    Coaxal Circles

    Conic Sections

    Note on the General Equation of the Second Degree

    The Chord of Contact of Tangents Drawn from an External Point to a Conic-Pole and Polar

    The Equation of a Pair of Tangents Drawn from an External Point to a Conic

    Conjugate Diameters

    The Polar Equation of a Conic

    Chapter XXI Coordinate Geometry of Three Dimensions

    The Coordinate System

    The Distance between Two Points

    The Coordinates of a Point Which Divides the Line Joining Two Given Points in a Given Ratio

    The Equation of a Plane

    The Equations of a Line—Direction Cosines

    The Angle between Two Directions

    The Intersection of Three Planes

    Answers to Exercises

    Index


Product details

  • No. of pages: 416
  • Language: English
  • Copyright: © Pergamon 1963
  • Published: January 1, 1963
  • Imprint: Pergamon
  • eBook ISBN: 9781483140889

About the Authors

C. Plumpton

W. A. Tomkys

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