MONOGRAPHS ON TOPICS OF MODERN MATHEMATICS

By MARY CLARK

CONTENTS WHAT MATHEMATICS IS WHY READ THIS BOOK? WHAT IS “DOING” MATHEMATICS? PROOFS FOUNDATIONS: SETS, RELATIONS, AND FUNCTIONS THE REAL NUMBERS FIELDS RINGS RING HOMOMORPHISMS PRIME FACTORIZATION ALGEBRAIC INTEGERS AND MODULES FERMAT’S LAST THEOREM ORDERED FIELDS THE LEAST UPPER BOUND AXIOM CONSTRUCTING THE REAL NUMBERS LIMITS AND CONTINUITY METRIC SPACES TOPOLOGICAL SPACES CLOSED SETS THE ZARISKI TOPOLOGY COMPACT SETS MORE ON COMPACTNESS SEQUENCES AND SERIES OF FUNCTIONS THE FUNDAMENTAL THEOREM OF ALGEBRA THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS 3 10 CONNECTEDNESS LINEAR ALGEBRA VECTOR SPACES BASES AND DIMENSION LINEAR TRANSFORMATIONS LINEAR SUBSPACES MATRICES THE DETERMINANT POSITIVE AND NEGATIVE VOLUME EVEN AND ODD PERMUTATIONS THE DETERMINANT OF A MATRIX TRANSPOSES AND PRODUCTS BACK TO LINEAR TRANSFORMATIONS THE CHARACTERISTIC POLYNOMIAL THE CAYLEY-HAMILTON THEOREM CANONICAL FORMS THE DERIVATIVE ASSUMPTIONS ON SCALARS A WEIRD VALUATION NORMED SPACES DE?NING THE DERIVATIVE THE DERIVATIVE’S SIGNI?CANCE THE CHAIN RULE PARTIAL DERIVATIVES COMPUTATION OF DERIVATIVES PRACTICAL COMPUTATION 6 10 ROLLE, CLAIRAUT, TAYLOR 6 11 DERIVATIVES OF SEQUENCES OF FUNCTIONS COMPLEX DI?ERENTIATION THE CAUCHY-RIEMANN EQUATIONS CONFORMAL MAPPINGS COMPLEX CLAIRAUT AND TAYLOR FUNCTIONS DE?NED BY POWER SERIES MULTIVARIATE POWER SERIES ANALYTIC CONTINUATION SMOOTH FUNCTIONS THE INVERSE FUNCTION THEOREM CURVED SPACE MANIFOLDS DI?ERENTIABLE MANIFOLDS ORIENTATION DI?ERENTIABLE FUNCTIONS THE TANGENT SPACE A COORDINATE-FREE DERIVATIVE THE REGULAR VALUE THEOREM GOING HIGHER DI?ERENTIAL GEOMETRY HYPERBOLIC SPACE CURVATURE SOME RIEMANN SURFACES THE FUNDAMENTAL GROUP CLASSI?CATION OF COMPACT MANIFOLDS 10 MORE AND MORE MATH 10 1 SOME OTHER BOOKS