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Ivanov O.A. — Easy as Pi?: An Introduction to Higher Mathematics
Ivanov O.A. — Easy as Pi?: An Introduction to Higher Mathematics

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Название: Easy as Pi?: An Introduction to Higher Mathematics

Автор: Ivanov O.A.

Аннотация:

This book aims at introducing the reader possessing some high school mathematics to both the higher and the more fundamental developments of the basic themes of elementary mathematics. To this end most chapters begin with a series of elementary problems, behind whose diverting formulation more advanced mathematical ideas lie hidden. These are then made explicit and further developments explored, thereby deepening and broadening the reader's understanding of mathematics - enabling him or her to see mathematics as a hologram. The book arose from a course for potential high school teachers of mathematics taught for several years at St. Petersburg University, and nearly every chapter ends with an interesting commentary on the relevance of its subject matter to the actual classroom setting. However, it can be recommended to a much wider readership, including university-level mathematics majors; even the professional mathematician will derive much pleasurable instruction from reading it.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 187

Добавлена в каталог: 02.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian group      83
Acceleration      130
Acceleration components      135
Acceleration, centripetal      135
Addition      4 6 155 164 167 169
Air resistance      152
Alexander the Great      10
Algebra with an identity element      121 123
Algebra without zero-divisors      121
Algebraic curve      69 79
Algebraic element      121
Algebraic extension      81
Algebraic number      81
Analysis      153
Analysis, foundations of      155
Analytic geometry      63
Arc, polygonal      66 96 137
Arithmetic-geometric mean inequality      46 51
Associative algebra      121 124
Associative law      5 119
Automorphism of an algebra      125
Autonomous system      108
Axiom of Archimedes      156 175
Axiom of Choice      170
Axiom of continuity      156
Axioms      5 156 158
Banach's fixed-point theorem      140
Basis of vector space      42 56 116 118 125
Bernoulli's inequality      15 50
Bernstein — Cantor — Schroeder theorem      99
Bezout's theorem      78
Bilinear map      113 121
Binomial coefficients      18 21 24
Binomial theorem      18 21
Bounded set      35 109 163
Bucephalus      11
Calculus      29 52 128 155
Cantor's theorem      145 163
Cantor's version of axiom of continuity      156
Cardinality      9 100
Cartesian coordinates      37 63 129
Cartesian product      5 10 13 119 121 124
Cauchy sequence      140 167
Cauchy's inequality      50 52
Cauchy's initial value problem      108
Cauchy's version of axiom of continuity      156
Cayley, theorem of      20
Celestial mechanics      135
Cell of a lattice      110
Central symmetry      33 40 56 111
Chain in a graph      89 99 100
Chain in a poset      170
Chain, rule      108 129
Characteristic of a field      82
Characteristic polynomial      23
Circular helix      134
Coefficient field      81
Combinations      16
Combinatorics      13
Commutative algebra      122
Commutative law      5
Complete space      140 156 166 168
Completion      166 173
complex numbers      75 76 81 118 122 124
Complex numbers in trigonometry      19
Components of acceleration      135
Composition of transformations      35
Congruence      110 113 114
Congruence, modulo $x^{k}$      25
Conjugate quaternion      120
Conjugate-transpose matrix      125
Connected component      66 89 94
Connected graph      89 90 92
Connected set      91
Conservative system      147
Continuity continuous curve      34
contour      147
Contracting map      140 141
Coordinate functions      128
Countable additivity      105
Critical edge of a graph      89 92
Critical point      144
Cross product      125 134 135 136
cube      34 119
Cubic curve      69
Curvature      136
Cycle in a graph      89
Cyclic group      82 114
Dangling edge      102
De Moivre's theorem      19
Dedekind cut      163
Dedekind's version of axiom of continuity      156 159 175
Degree of a point      35
Degree of a vertex      87
Derivative      18 59 68 128 134
Derivative, partial      108 141
Determinant      74 76 80 107 110 116 119
Diameter of a set      129
Difference between cuts      165
Differentiability      133 142
Differential equations      107 133 136
Dilation      111 135
Dimension of a vector space      81 82 122
Dirichlet      103
Discriminant      75
Division algebra      121
Division algorithm      77 81 82
dodecahedron      94
Dot product      125 129
Edge of a graph      87
Edge, critical      89
Eigenvalues      23
Elimination      80
ellipse      61 65 80 135 145
Energy      147
Equation Diophantine      66 113
Equation of a set      34 64
Equilibrium, stable      149
Equilibrium, unstable      152
EQUIVALENCE      3 155 167 168
Equivalence class      3 155 167 168
Equivalent norms      56 173
Euclidean metric      56
Euclidean motion      37 125
Euclidean motion, discrete      43
Euclidean space      37 55 118 121 125 128
Euler      114
Euler characteristic      94
Euler's formula for a planar graph      92
Euler's identity      26 116 120
Euler's method of approximate solution of a d.e.      137
Eulerian graph      100
Extreme-value theorem      52 56 163
Factor of a polynomial      24 48
Factor Theorem      68 70
Fermat — Euler theorem      113
Fermat's great theorem      67
Fermat's last theorem      67
Fermat's little theorem      113
Fermat's principle      131
Fermat's theorem      129
Fibonacci numbers      21 105
Field      122
Field extension      80 83 122
Field of complex numbers      81
Field of fractions      155
Field of real numbers      156
Field, finite      82
Field, ordered, Archimedean      156 168
Filter      169
Finite field      82
Finite graph      87
Finite hyperreal number      159
Finite set      9
Finite-dimensional algebra      121 124
Fixed point      140
Formal logic      160
Fourier coefficients      60
Fractional part of a real number      64
Friction      152
Frieze pattern      39
Frobenius' Theorem      123
Fubini's theorem      29 115
Full order      156
Function, concave down      53
Function, continuous      53 66 107 134
Function, smooth      142
Function, vector-valued      128
Functional equation      133
Fundamental region      110
Fundamental theorem of algebra      122
Gamma function      29
General lattice      43
Generating function      18 21
Generators of a group      38 41
Geometric series      140
Geometric transformation      32
Graph      87
Graph of an equation      64
Graph, bipartite      98
Graph, connected      89
Graph, planar      92
Graph, simple unoriented      87
Gravitational field      135 148
Green's theorem      61
Group      38 40 43 82 83 107 113
Group of rigid motions      38 125
Group, multiplicative      82
Hall's theorem      98
Hamilton      120
hilbert      95
Hoelder's inequality      54
Homogeneous function      57
Homogeneous linear system      70 119
Homothety      135
Hyperbola      151
Hyperreal field      159
icosahedron      94
Implicit function theorem      154
Inclusion-exclusion formula      13
Incommensurable numbers      35 106
Index of a point      96
Index of a ray      35
Inequality      46
Inequality between the arithmetic and geometric means      46
Infinite decimal      166
Infinitely close elements      159
Infinitely large element      158
Infinitesimal element      157 171
Inner product      59
Integers      3
Integral domain      82 155
Integral equation      138
Integral of a differential equation      147
Integration by parts      29
Interior of a subset      56
Intermediate-value theorem      66 139 163
Intersection      63 161
Interval method      65
Intrinsic subset      161
Intuitionist      170
Inverse-function theorem      143
Irrational numbers      106 107
Irreducible polynomial      81
Isometry      125
Isoperimetric problem      61
Isoperimetric property of circle      62
Jacobian      109 116
Jacobian matrix      108 143 145
Jensen's inequality      57
Jordan curve theorem      94
Jordan form of a matrix      23
Kepler's laws of motion      135
Klein      xiii
Koenigsberg bridge problem      100
Lagrange's four-squares theorem      113
Lagrange's theorem in group theory      113
Latin rectangle      99
Lattice of points      43 110 114
Law of conservation of energy      147
Least upper bound property      157
Lebesgue integral      59
Lebesgue measure      106 144
Leibniz' rule      150
Level curve      147 148
LIMIT      28 52 107 128 133 134 138 140 142 151 156 162 167 174
Linear equations      68 70 75 76 119
Linear independence      82 121 122
Linear transformation      37 121 124
Linearization      142
Liouville's theorem      109
Locally volume-preserving map      110
Lower number of a cut      164
Marriage problem      98
Mathematical induction      1 8
Mathematical logic      160 171
Matrix algebra      123
Matrix multiplication      118
Matrix of a linear transformation      124
Mean-value theorem      52 129 141
Means of three numbers      48
Means of two numbers      47
Measure      58 105
Measure-preserving map      105 109
Method of elimination      80
Method of mathematical induction      8
Metric space      139
Minimal polynomial      81
Minkowski's inequality      54
Minkowski's Lemma      111
Minkowski's norm      56
Morse — Sard theorem      144
Motion in a gravitational field      135
Motion of a point-particle      134
Motion of a rigid rod      130
Multilinear map      119
Multiple integral      29 115
Multiple root      68 75 144
Multiplicative group of a field      113
Natural numbers      6
Newton's Second Law of Motion      136 148
Newton's universal law of gravitation      135
Nonarchimedean field      158
NonArchimedean norm      173
Nondegenerate critical point      149
Nonstandard analogue      159
Nonstandard line      158
Nonstandard line, model of      168
Nonstandard proof      163
Nonstandard sequence      161
Nonstandard statement      163
Norm of a quaternion      120
Norm on a vector space      50 55
Norm on the rationals      172
Number of elements of a set      9
Number of partitions of a positive integer      24
Number, $\pi$      28 29 115
Number, e      28 137
Number, n! (n-factorial)      14 28
octahedron      94
1 2
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