Towards Higher Mathematics: A CompanionContaining a large and varied set of problems, this rich resource will allow students to stretch their mathematical abilities beyond the school syllabus, and bridge the gap to university-level mathematics. Many proofs are provided to better equip students for the transition to university. The author covers substantial extension material using the language of sixth form mathematics, thus enabling students to understand the more complex material. Exercises are carefully chosen to introduce students to some central ideas, without building up large amounts of abstract technology. There are over 1500 carefully graded exercises, with hints included in the text, and solutions available online. Historical and contextual asides highlight each area of mathematics and show how it has developed over time. |
Contents
| 1 | |
| 8 | |
2 Induction | 68 |
3 Vectors and Matrices | 128 |
4 More on Matrices | 282 |
5 Techniques of Integration | 341 |
6 Differential Equations | 426 |
Hints to Selected Exercises | 490 |
| 515 | |
| 517 | |
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Common terms and phrases
algebraic angle calculate circle co-ordinates coefficients column complex numbers consider constant Corollary curve Deduce defined Definition denote detA Determine diagonal diagonalizable differential eigenvalues eigenvectors elementary matrices ellipse entries equals equation EROS Example Exercises Figure Find formula given graph Hence hyperbolic identity independent induction infinite initial-value problem integer integral invertible invertible matrix isometry Laplace transform linear combination linear map linear system mathematician mathematics Möbius transformation multiplication natural number non-zero notation Note orthogonal matrix orthonormal basis parametric particular solution permutation matrix plane polynomial positive integer prime numbers Proof Proposition prove real numbers recurrence relation roots RRE form RRE(A satisfies scalar sequence Show sinx solve square matrix subset subspace substitution theorem trapezium rule triangle v₁ values variables vectors Verify write zero