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Contents Preface I Problems 1 Real Analysis 1.1 Elementary Calculus 1.2 Limitsand Continuity 1.3 Sequences, Series, and Products 1.4 Differential Calculus 1.5 Integral Calculus 1.6 Sequences of Functions 1.7 Fourier Series 1.8 Convex Functions 2 Multivariable Calculus 2.1 Limitsand Continuity 2.2 Differential Calculus 2.3 Integral Calculus 3 Differential Equations 3.1 First Order Equations 3.2 SecondOrder Equations 3.3 Higher Order Equations 3.4 Systems of Differential Equations 4 Metric Spaces 4.1 Topology of Rn 4.2 General Theory 4.3 Fixed Point Theorem 5 Complex Analysis 5.1 Complex Numbers 5.2 Series and Sequences of Functions 5.3 Conformal Mappings 5.4 Functions on the Unit Disc 5.5 Growth Conditions 5.6 Analytic and Meromorphic Functions 5.7 Cauchy’s Theorem 5.8 Zeros and Singularities 5.9 Harmonic Functions 5.10 Residue Theory 5.11 Integrals Along the Real Axis 6 Algebra 6.1 Examples of Groups and General Theory 6.2 Homomorphisms and Subgroups 6.3 Cyclic Groups 6.4 Normality, Quotients, and Homomorphisms 6.5 Sn, An , Dn, .. 6.6 Direct Products 6.7 Free Groups, Generators, and Relations 6.8 Finite Groups 6.9 Ringsand Their Homomorphisms 6.10 Ideals 6.11 Polynomials 6.12 Fields and Their Extensions 6.13 Elementary Number Theory 7 Linear Algebra 7.1 Vector Spaces 7.2 Rankand Determinants 7.3 Systems of Equations 7.4 Linear Transformations 7.5 Eigenvalues and Eigenvectors 7.6 Canonical Forms 7.7 Similarity 7.8 Bilinear, Quadratic Forms, and Inner Product Spaces 7.9 General Theory ofMatrices II Solutions 1 Real Analysis 1.1 Elementary Calculus 1.2 Limits and Continuity 1.3 Sequences, Series, and Products 1.4 Differential Calculus 1.5 Integral Calculus 1.6Sequences of Functions 1.7 Fourier Series 1.8 Convex Functions 2 Multivariable Calculus 2.1 Limitsand Continuity 2.2 Differential Calculus 2.3 Integral Calculus 3 Differential Equations 3.1 First Order Equations 3.2 Second Order Equations 3.3 Higher Order Equations 3.4 Systems of Differential Equations 4 Metric Spaces 4.1 Topology of Rn 4.2 General Theory 4.3 Fixed Point Theorem 5 Complex Analysis 5.1 Complex Numbers 5.2 Series and Sequences of Functions 5.3 Conformal Mappings 5.4 Functions on the Unit Disc 5.5 Growth Conditions 5.6 Analytic and Meromorphic Functions 5.7 Cauchy’s Theorem 5.8 Zeros and Singularities 5.9 Harmonic Functions 5.10 Residue Theory 5.11 Integrals Along the Real Axis 6 Algebra 6.1 Examples of Groups and General Theory 6.2 Homomorphisms and Subgroups 6.3 Cyclic Groups 6.4 Normality, Quotients, and Homomorphisms 6.5 Sn, An , Dn, .. 6.6 Direct Products 6.7 Free Groups, Generators, and Relations 6.8 Finite Groups 6.9 Rings and Their Homomorphisms 6.10 Ideals 6.11 Polynomials 6.12 Fields and Their Extensions 6.13 Elementary Number Theory 7 Linear Algebra 7.1 Vector Spaces 7.2 Rankand Determinants 7.3 Systems of Equations 7.4 Linear Transformations 7.5 Eigenvalues and Eigenvectors 7.6 Canonical Forms 7.7 Similarity 7.8 Bilinear, Quadratic Forms, and Inner Product Spaces 7.9 General Theory of Matrices III Appendices A How to Get the Exams A.1 On-line A.2 Off-line, the Last Resort B Passing Scores C The Syllabus References Index