Plumpton / Tomkys Sixth Form Pure Mathematics

Volume 2
1. Auflage 2014
ISBN: 978-1-4831-4088-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 416 Seiten, Web PDF

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.

Autoren/Hrsg.

Plumpton, C.

Tomkys, W. A.

Inhaltsverzeichnis

1;Front Cover
;1
2;Sixth Form Pure Mathematics;4
3;Copyright Page;5
4;Table of Contens;6
5;PREFACE;9
6;CHAPTER XI. LINEAR EQUATIONS AND DETERMINANTS;10
6.1;11.1 The solution of linear simultaneous equations in two unknowns;10
6.2;11.2 Second order determinants;17
6.3;11.3 Rules of manipulation for second order determinants;18
6.4;11.4 Third order determinants;20
6.5;11.5 Factorization o! determinants;24
6.6;11.6 Geometrical interpretation;27
6.7;11.7 The equation of a straight line through two given points;28
6.8;11.8 The equation of a tangent to the curve;30
6.9;11.9 The equation of a line-pair;32
6.10;11.10 The equation of a circle through three given points;33
6.11;11.11 The area of a triangle;34
6.12;11.12 The solution of simultaneous equations;36
6.13;11.13 Summary;38
6.14;11.14 The product of two determinants;39
6.15;11.15 The derivative of a determinant;39
7;CHAPTER XIIINVERSE CIRCULAR FUNCTIONS, HYPERBOLICFUNCTIONS AND INVERSE HYPERBOLIC FUNCTIONS;50
7.1;12.1 Inverse circular functions;50
7.2;12.2 The derivatives of the inverse circular functions;53
7.3;12.3 Standard integrals;56
7.4;12.4 The hyperbolic functions;59
7.5;12.5 Inverse hyperbolic functions;67
7.6;12.6 Derivatives of the inverse hyperbolic functions;68
7.7;12.7 Logarithmic forms of the inverse hyperbolic functions;70
7.8;12.8 Methods of integration;71
7.9;12.9 The Integrals;73
7.10;12.10 Summary of standard integrals and methods of integration so farconsidered;75
8;CHAPTER XIII. DEFINITE INTEGRALSFURTHER APPLICATIONS OF INTEGRATION;84
8.1;13.1 Properties of definite integrals;84
8.2;13.2 Infinite integrals;90
8.3;13.3 Reduction formulae;96
8.4;13.4 Approximate numerical integration;102
8.5;13.5 Mean values and root mean square;110
8.6;13.6 Centre of mass;114
8.7;13.7 The theorem of Pappus concerning volumes;117
8.8;13.8 Moments of inertia;121
9;CHAPTER XIV. SOME PROPERTIES OF CURVES;138
9.1;14.1 Points of inflexion;138
9.2;14.2 The length of a curve;144
9.3;14.3 The cycloid;149
9.4;14.4 Areas of surfaces of revolution;151
9.5;14.5 The theorem of Pappus concerning surfaces of revolution;154
9.6;14.6 Curvature;157
9.7;14.7 Newton's formula for radius of curvature at the origin;162
10;CHAPTER XV. POLAR COORDINATES;170
10.1;15.1 Definitions;170
10.2;15.2 Loci in polar coordinates;171
10.3;15.3 Curve sketching in polar coordinates;175
10.4;15.4 The lengths of chords of polar curves which are drawn through thepole;179
10.5;15.5 Transformations from polar to cartesian equations and the reverseprocess;181
10.6;15.6 Areas in polar coordinates;183
10.7;15.7 The length of an arc in polar coordinates;187
10.8;15.8 Volumes of revolution and areas of surfaces of revolution in polarcoordinates;188
10.9;15.9 The angle between the tangent and the radius vector;192
10.10;15.10 The tangential polar equation—Curvature;194
11;CHAPTER XVI. COMPLEX NUMBERS;202
11.1;16.1 The number system;202
11.2;16.2 Definition of complex number;202
11.3;16.3 The cube roots of unity;206
11.4;16.4 Conjugate pairs of complex roots;208
11.5;16.5 The geometry of complex numbers;210
11.6;16.6 The polar coordinate form of a complex number—Modulus andArgument;214
11.7;16.7 Products and quotients;216
11.8;16.8 De Moivre's Theorem;223
11.9;16.9 The exponential form of a complex number;230
11.10;16.10 Exponential values of sine and cosine;232
12;CHAPTER, XVII. DIFFERENTIAL EQUATIONS;246
12.1;17.1 Formation 01 differential equations;246
12.2;17.2 The solution of a differential equation;249
12.3;17.3 First order differential equations with variables separable;251
12.4;17.4 Homogeneous equations;253
12.5;17.5 The law of natural growth;255
12.6;17.6 Linear equations of the first order;261
12.7;17.8 Equations of higher orders;266
12.8;17.9 Linear equations of the second order with constant coefficients;268
12.9;17.10 The complementary function;270
12.10;17.11 The particular integral;273
13;CHAPTER XVIII. APPROXIMATE NUMERICAL SOLUTION OFEQUATIONS;289
13.1;18.1 Graphical methods;289
13.2;18.2 The number of real roots of an equation;292
13.3;18.3 The approximate value of a small root of a polynomial equation;295
13.4;18.4 Newton's method for obtaining a closer approximation to a realroot of an equation;297
14;CHAPTER XIX. INEQUALITIES;306
14.1;19.1 Rules of manipulation;306
14.2;19.2 Fundamental inequalities;315
14.3;19.3 The calculus applied to inequalities;318
15;CHAPTER XX. COORDINATE GEOMETRY;325
15.1;20.1 The straight line;325
15.2;20.2 Line pairs;327
15.3;20.3 The circle;328
15.4;20.4 The radical axis;330
15.5;20.5 Coaxal circles;332
15.6;20.6 Conic sections;337
15.7;20.7 Note on the general equation of the second degree;337
15.8;20.8 The chord of contact of tangents drawn from an external point toa conic. Pole and polar;338
15.9;20.9 The equation of a pair of tangents drawn from an external pointto a conic;339
15.10;20.10 Conjugate diameters;343
15.11;20.11 The polar equation of a conic;346
16;CHAPTER XXI. COORDINATE GEOMETRY OF THREE DIMENSIONS;360
16.1;21.1 The coordinate system;360
16.2;21.2 The distance between two points;362
16.3;21.3 The coordinates o! a point which divides the line joining two givenpoints in a given ratio;362
16.4;21.4 The equation of a plane;364
16.5;21.5 The equations of a line—Direction Cosines;365
16.6;21.6 The angle between two directions;372
17;ANSWERS TO EXERCISES;392
18;SUBJECT INDEX;414