Further Pure Mathematics-Oxford
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English
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Entries: 126
Brian Gaulter
Mark Gaulter
pure math
oxford
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Further Pure Mathematics-Oxford
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126
Preface
Other
Page 5
1. Complex Numbers
Theory
Page 7
1A. What is a complex number?
Theory
Page 7
1B. Calculating with complex numbers
Theory
Page 8
1C. Argand diagram
Theory
Page 12
1D. Loci in the complex plane
Theory
Page 18
1E. Cube roots of unity
Theory
Page 24
2. Further Trigonometry with Calculus
Theory
Page 28
2A. General solutions of trigonometric equations
Theory
Page 28
2B. Harmonic form
Theory
Page 32
2C. Inverse trigonometric functions
Theory
Page 36
3. Polar Coordinates
Theory
Page 49
3A. Position of a point
Theory
Page 49
3B. Connection between polar and cartesian coordinates
Theory
Page 50
3C. Sketching curves given in polar coordinates
Theory
Page 51
3D. Area of a sector of a curve
Theory
Page 54
3E. Equations of the tangents to a curve
Theory
Page 59
4. Differential Equations
Theory
Page 63
4A. First-order equations requiring an integrating factor
Theory
Page 63
4B. Second-order differential equations
Theory
Page 67
4C. Solution of differential equations by substitution
Theory
Page 81
5. Determinants
Theory
Page 86
5A. Definition of 2 × 2 and 3 × 3 determinants
Theory
Page 86
5B. Rules for the manipulation of determinants
Theory
Page 87
5C. Factorisation of determinants
Theory
Page 90
5D. Solution of three equations in three unknowns
Theory
Page 93
6. Vector Geometry
Theory
Page 100
6A. Vector equation of a line
Theory
Page 100
6B. Cartesian equation of a line
Theory
Page 101
6C. Resolved part of a vector
Theory
Page 104
6D. Direction ratios
Theory
Page 104
6E. Direction cosines
Theory
Page 105
6F. Vector product
Theory
Page 108
6G. Area of a triangle
Theory
Page 111
6H. Equation of a plane
Theory
Page 112
6I. Distance of a plane from the origin
Theory
Page 118
6J. Distance of a plane from a point
Theory
Page 119
6K. Scalar triple product and its applications
Theory
Page 127
7. Curve Sketching and Inequalities
Theory
Page 136
7A. Curve sketching
Theory
Page 136
7B. Sketching rational functions with a quadratic denominator
Theory
Page 139
7C. Inequalities
Theory
Page 146
8. Roots of Polynomial Equations
Theory
Page 153
8A. Roots of a quadratic equation
Theory
Page 153
8B. Roots of a cubic equation
Theory
Page 155
8C. Roots of a polynomial equation of degree n
Theory
Page 156
8D. Equations with related roots
Theory
Page 158
8E. Complex roots of a polynomial equation
Theory
Page 160
9. Proof, Sequences and Series
Theory
Page 165
9A. Proof by induction
Theory
Page 165
9B. Proof by contradiction
Theory
Page 170
9C. Summation of series
Theory
Page 174
9D. Convergence
Theory
Page 181
9E. Maclaurin's series
Theory
Page 183
9F. Using power series
Theory
Page 188
9G. Power series for more complicated functions
Theory
Page 191
10. Hyperbolic Functions
Theory
Page 195
10A. Definitions
Theory
Page 195
10B. Graphs of cosh x, sinh x and tanh x
Theory
Page 196
10C. Standard hyperbolic identities
Theory
Page 197
10D. Differentiation of hyperbolic functions
Theory
Page 198
10E. Integration of hyperbolic functions
Theory
Page 199
10F. Inverse hyperbolic functions
Theory
Page 200
10G. Logarithmic form
Theory
Page 204
10H. Differentiation of inverse hyperbolic functions
Theory
Page 207
10I. Double-angle formulae
Theory
Page 216
10J. Power series
Theory
Page 218
10K. Osborn's rule
Theory
Page 219
11. Conics
Theory
Page 222
11A. Generating conics
Theory
Page 222
11B. Parabola
Theory
Page 223
11D. Hyperbola
Theory
Page 226
12. Further Integration
Theory
Page 231
12A. Inverse function of a function rule
Theory
Page 231
12B. Integration by parts
Theory
Page 232
12C. Integration of fractions
Theory
Page 233
12D. Reduction formulae
Theory
Page 236
12E. Arc length
Theory
Page 242
12G. Improper integrals
Theory
Page 246
12H. Summation of series
Theory
Page 248
13. Numerical Methods
Theory
Page 251
13A. Solution of polynomial equations
Theory
Page 258
13B. Evaluation of areas under curves
Theory
Page 261
13C. Step-by-step solution of differential equations
Theory
Page 267
13D. Taylor's series
Theory
Page 269
14. Matrices
Theory
Page 273
14A. Notation
Theory
Page 273
14B. The order of a matrix
Theory
Page 273
14C. Addition and subtraction of matrices
Theory
Page 274
14D. Multiplication of matrices
Theory
Page 274
14E. Determinant of a matrix
Theory
Page 277
14F. Identity matrices and zero matrices
Theory
Page 277
14G. Inverse matrices
Theory
Page 278
14H. Transformations
Theory
Page 281
14I. Eigenvectors and eigenvalues
Theory
Page 283
14J. Diagonalisation
Theory
Page 285
14K. The characteristic equation
Theory
Page 288
15. Further Complex Numbers
Theory
Page 292
15A. De Moivre's theorem
Theory
Page 292
15B. nth roots of unity
Theory
Page 296
15C. Exponential form of a complex number
Theory
Page 299
15D. Trigonometric identities
Theory
Page 302
15E. Transformations in a complex plane
Theory
Page 310
16. Intrinsic Coordinates
Theory
Page 315
16A. Trigonometric functions of ψ
Theory
Page 315
16B. Radius of curvature
Theory
Page 316
16C. Finding intrinsic equations
Theory
Page 317
17. Groups
Theory
Page 321
17A. Binary and unary operations
Theory
Page 375
17B. Modular arithmetic
Theory
Page 322
17C. Definition of a group
Theory
Page 322
17D. Group table
Theory
Page 324
17E. Symmetries of a regular n-sided polygon
Theory
Page 325
17F. Non-finite groups
Theory
Page 326
17G. a^m notation
Theory
Page 327
17H. Permutation groups
Theory
Page 327
17I. Generator of a group
Theory
Page 328
17J. Cyclic groups
Theory
Page 328
17K. Abelian groups
Theory
Page 329
17L. Order of a group
Theory
Page 332
17O. Isomorphic groups
Theory
Page 333
17Q. Groups of order 3
Theory
Page 334
17R. Groups of order 4
Theory
Page 334
17T. Groups of order 6
Theory
Page 335
17U. Real vector spaces
Theory
Page 339
Answers
Answer
Page 342
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