Contents
Preface vii
I Problems 1
1 Real Analysis 3
1.1 Elementary Calculus 3
1.2 Limits and Continuity 8
1.3 Sequences,Series,and Products 10
1.4 Differential Calculus 14
1.5 Integral Calculus 18
1.6 Sequences of Functions 22
1.7 Fourier Series 27
1.8 Convex Functions 29
2 Multivariable Calculus 31
2.1 Limits and Continuity 31
2.2 Differential Calculus 32
2.3 Integral Calculus 40
3 Differential Equations 43
3.1 First Order Equations 43
3.2 Second Order Equations 47
3.3 Higher Order Equations 49
3.4 Systems of Differential Equations 50
4 Metric Spaces 57
4.1 Topology of Rn 57
4.2 General Theory 60
5 Complex Analysis 65
5.1 Complex Numbers 65
5.2 Series and Sequences of Functions 67
5.3 Conformal Mappings 70
5.4 Functions on the Unit Disc 71
5.5 Growth Conditions 74
5.6 Analytic and Meromorphic Functions 75
5.7 Cauchy's Theorem 80
5.8 Zeros and Singularities 82
5.9 Harmonic Functions 86
5.10 Residue Theory 87
5.11 Integrals Alongthe Real Axis 93
6 Algebra 97
6.1 Examples of Groups and General Theory 97
6.2 Homomorphisms and Subgroups 99
6.4 Normality,Quotients,and Homomorphisms 102
6.7 Free Groups,Generators,and Relations 106
6.9 Rings and Their Homomorphisms 109
6.12 Fields and Their Extensions 116
6.13 Elementary Number Theory 118
7 Linear Algebra 123
7.1 Vector Spaces 123
7.2 Rank and Determinants 125
7.3 Systems of Equations 129
7.4 Linear Transformations. 129
7.5 Eigenvalues and Eigenvectors 134
7.8 Bilinear,Quadratic Forms,and lnner Product Spaces 146
7.9 General Theory of Matrices 149
II Solutions 155
1 Real Analysis 157
1.1 Elementary Calculus 157
1.2 Limitsand Continuity 173
1.3 Sequences,Series,and Products 178
1.4 Differential Calculus 191
1.5 Integral Calculus 200
1.6 Sequences of Functions 213
1.7 Fourier Series 228
1.8 Convex Functions 232
2 Multivariable Calculus 235
2.1 Limitsand Continuity 235
2.2 Differential Calculus 237
2.3 Integral Calculus 258
3 Differential Equations 265
3.1 First Order Equations 265
3.2 Second Order Equations 274
3.3 Higher Order Equations 278
3.4 Systems of Differential Equations 280
4 Metric Spaces 289
4.1 Topology of Rn 289
4.2 General Theory 297
4.3 Fixed Point Theorem 300
5 Complex Analysis 305
5.1 Complex Numbers 305
5.2 Series and Sequences of Functions 309
5.3 Conformal Mappings 316
5.4 Functionson the Unit Disc 321
5.5 Growth Conditions 329
5.6 Analytic and Meromorphic Functions 334
5.7 Cauchy's Theorem 347
5.8 Zeros and Singularities 356
5.9 Harmonic Functions 372
5.10 Residue Theory 373
5.11 Integrals Along the Real Axis 390
6 Algebra 423
6.1 Examples of Groups and General Theory 423
6.2 Homomorphisms and Subgroups 429
6.3 Cyclic Groups 433
6.4 Normality,Quotients,and Homomorphisms 435
6.5 Sn,An,Dn,… 440
6.6 Direct Products 443
6.7 Free Groups,Generators,and Relations 445
6.8 Finite Groups 450
6.9 Rings and Their Homomorphisms 456
6.10 Ideals 460
6.11 Polynomials 463
6.12 Fields and Their Extensions 473
6.13 Elementary Number Theory 480
7 Linear Algebra 489
7.1 Vector Spaces 489
7.2 Rank and Determinants 495
7.3 Systems of Equations 501
7.4 Linear Transformations 503
7.5 Eigenvaluesand Eigenvectors 514
7.6 Canonical Forms 525
7.7 Similarity 540
7.8 Bilinear,Quadratic Forms,and lnner Product Spaces 545
7.9 General Theory of Matrices 553
Ⅲ Appendices 569
A How to Get the Exams 571
A.1 On-line 571
A.2 0ff-Iine,the Last Resort 571
B Passing Scores 577
C The Syllabus 579
References 581
Index 589