## All the Mathematics You Missed: But Need to Know for Graduate SchoolFew beginning graduate students in mathematics and other quantitative subjects possess the daunting breadth of mathematical knowledge expected of them when they begin their studies. This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, point-set topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations. |

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This is a fabulous book. It's chapters are logically organized, extremely accessible, actually enjoyable, and provide just what TG intends. Sure, for each topic there's much more one could describe, and for each one could take a different tack, but in any intro graduate course the prof will take their own trajectory, anyway.

Lovely.

User Review - Flag as inappropriate

奇書一本。當初於中大Swindon見此書，研讀月餘，然及後閒來再讀總覺有新得著。此書非為深入了解，但勝在精博，各方數學科目總有涉及。尤適合我輩在大學中未有下苦功讀數，後有悔忱之人。

### Contents

Preface | xiii |

On the Structure of Mathematics | xix |

Brief Summaries of Topics | xxiii |

04 Point Set Topology | xxiv |

08 Geometry | xxv |

010 Countability and the Axiom of Choice | xxvi |

014 Differential Equations | xxvii |

Linear Algebra | 1 |

Curvature for Curves and Surfaces | 145 |

72 Space Curves | 148 |

73 Surfaces | 152 |

74 The GaussBonnet Theorem | 157 |

75 Books | 158 |

Geometry | 161 |

81 Euclidean Geometry | 162 |

82 Hyperbolic Geometry | 163 |

12 The Basic Vector Space Rⁿ | 2 |

13 Vector Spaces and Linear Transformations | 4 |

14 Bases Dimension and Linear Transformations as Matrices | 6 |

15 The Determinant | 9 |

16 The Key Theorem of Linear Algebra | 12 |

17 Similar Matrices | 14 |

18 Eigenvalues and Eigenvectors | 15 |

19 Dual Vector Spaces | 20 |

110 Books | 21 |

ϵ and 𝛿 Real Analysis | 23 |

22 Continuity | 25 |

23 Differentiation | 26 |

24 Integration | 28 |

25 The Fundamental Theorem of Calculus | 31 |

26 Pointwise Convergence of Functions | 35 |

27 Uniform Convergence | 36 |

28 The Weierstrass MTest | 38 |

29 Weierstrass Example | 40 |

210 Books | 43 |

211 Exercises | 44 |

Calculus for VectorValued Functions | 47 |

32 Limits and Continuity of VectorValued Functions | 49 |

33 Differentiation and Jacobians | 50 |

34 The Inverse Function Theorem | 53 |

35 Implicit Function Theorem | 56 |

36 Books | 60 |

Point Set Topology | 63 |

42 The Standard Topology on Rⁿ | 66 |

43 Metric Spaces | 72 |

44 Bases for Topologies | 73 |

45 Zariski Topology of Commutative Rings | 75 |

46 Books | 77 |

47 Exercises | 78 |

Classical Stokes Theorems | 81 |

51 Preliminaries about Vector Calculus | 82 |

512 Manifolds and Boundaries | 84 |

513 Path Integrals | 87 |

514 Surface Integrals | 91 |

515 The Gradient | 93 |

517 The Curl | 94 |

52 The Divergence Theorem and Stokes Theorem | 95 |

53 Physical Interpretation of the Divergence Thm | 97 |

54 A Physical Interpretation of Stokes Theorem | 98 |

55 Proof of the Divergence Theorem | 99 |

56 Sketch of a Proof for Stokes Theorem | 104 |

57 Books | 108 |

Differential Forms and Stokes Theorem | 111 |

61 Volumes of Parallelepipeds | 112 |

62 Diff Forms and the Exterior Derivative | 115 |

622 The Vector Space of 𝓀forms | 118 |

623 Rules for Manipulating 𝓀forms | 119 |

624 Differential 𝓀forms and the Exterior Derivative | 122 |

63 Differential Forms and Vector Fields | 124 |

64 Manifolds | 126 |

65 Tangent Spaces and Orientations | 132 |

652 Tangent Spaces for Abstract Manifolds | 133 |

653 Orientation of a Vector Space | 135 |

654 Orientation of a Manifold and its Boundary | 136 |

66 Integration on Manifolds | 137 |

67 Stokes Theorem | 139 |

68 Books | 142 |

69 Exercises | 143 |

83 Elliptic Geometry | 166 |

84 Curvature | 167 |

85 Books | 168 |

86 Exercises | 169 |

Complex Analysis | 171 |

91 Analyticity as a Limit | 172 |

92 CauchyRiemann Equations | 174 |

93 Integral Representations of Functions | 179 |

94 Analytic Functions as Power Series | 187 |

95 Conformal Maps | 191 |

96 The Riemann Mapping Theorem | 194 |

Hartogs Theorem | 196 |

98 Books | 197 |

99 Exercises | 198 |

Countability and the Axiom of Choice | 201 |

102 Naive Set Theory and Paradoxes | 205 |

103 The Axiom of Choice | 207 |

104 Nonmeasurable Sets | 208 |

105 Gödel and Independence Proofs | 210 |

106 Books | 211 |

Algebra | 213 |

112 Representation Theory | 219 |

113 Rings | 221 |

114 Fields and Galois Theory | 223 |

115 Books | 228 |

116 Exercises | 229 |

Lebesgue Integration | 231 |

122 The Cantor Set | 234 |

123 Lebesgue Integration | 236 |

124 Convergence Theorems | 239 |

125 Books | 241 |

Fourier Analysis | 243 |

132 Fourier Series | 244 |

133 Convergence Issues | 250 |

134 Fourier Integrals and Transforms | 252 |

135 Solving Differential Equations | 256 |

136 Books | 258 |

Differential Equations | 261 |

142 Ordinary Differential Equations | 262 |

1431 Mean Value Principle | 266 |

1432 Separation of Variables | 267 |

1433 Applications to Complex Analysis | 270 |

1451 Derivation | 273 |

1452 Change of Variables | 277 |

Integrability Conditions | 279 |

147 Lewys Example | 281 |

148 Books | 282 |

Combinatorics and Probability Theory | 285 |

152 Basic Probability Theory | 287 |

153 Independence | 290 |

154 Expected Values and Variance | 291 |

155 Central Limit Theorem | 294 |

156 Stirlings Approximation for 𝑛 | 300 |

157 Books | 305 |

Chapter 16 Algorithms | 307 |

161 Algorithms and Complexity | 308 |

163 Sorting and Trees | 313 |

164 PNP? | 316 |

Newtons Method | 317 |

166 Books | 324 |

### Other editions - View all

All the Mathematics You Missed: But Need to Know for Graduate School Thomas A. Garrity Limited preview - 2002 |

All the Mathematics You Missed: But Need to Know for Graduate School Thomas A. Garrity Limited preview - 2001 |

### Common terms and phrases

1-forms algorithm analytic functions Axiom of Choice ball basic basis boundary Cauchy-Riemann equations chapter circle coefficients column vector complex analysis complex numbers compute continuous function converges uniformly coordinates countable curve define definition denoted differentiable functions differential equations Divergence Theorem eigenvalues elements elliptic geometry example exists exterior derivative fc-forms finite Fourier series Fourier transform function f(x Fundamental Theorem given goal graph hence holomorphic infinite interval intuitions inverse Jacobian Lebesgue integral Lemma length limit linear algebra linear transformation loop manifold mathematics means measure normal vector Note notion open set orientation parametrization path integral permutation plane polynomial problem proof rational numbers real numbers real-valued function rectangles roots solution Stokes subgroup subset surface tangent line tangent vector Theorem of Calculus tion topological space uniform convergence vector field vector space vector-valued zero