Berkeley Problems in Mathematics

I Problems.- 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.- 1.6 Sequences of Functions.- 1.7 Fourier Series.- 1.8 Convex Functions.- 2 Multivariable Calculus.- 2.1 Limits and 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 Integral Representation of Analytic Functions.- 5.5 Functions on the Unit Disc.- 5.6 Growth Conditions.- 5.7 Analytic and Meromorphic Functions.- 5.8 Cauchy's Theorem.- 5.9 Zeros and Singularities.- 5.10 Harmonic Functions.- 5.11 Residue Theory.- 5.12 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, Products, 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 Rank and 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.- 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.- 1.6 Sequences of Functions.- 1.7 Fourier Series.- 1.8 Convex Functions.- 2 Multivariable Calculus.- 2.1 Limits and 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 Integral Representation of Analytic Functions.- 5.5 Functions on the Unit Disc.- 5.6 Growth Conditions.- 5.7 Analytic and Meromorphic Functions.- 5.8 Cauchy's Theorem.- 5.9 Zeros and Singularities.- 5.10 Harmonic Functions.- 5.11 Residue Theory.- 5.12 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, Products, 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 Rank and 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.- Appendices.