Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds

Description

FineReader OCR CONTENTS PREFACE 1 VECTORS AND MATRICES 1 VECTORS IN RN 2 DOT PRODUCT 3 SUBSPACES OF R" 4 LINEAR TRANSFORMATIONS AND MATRIX ALGEBRA 5 INTRODUCTION TO DETERMINANTSAND THE CROSS PRODUCT 2 FUNCTIONS, LIMITS,AND CONTINUITY 1 SCALAR-AND VECTOR-VALUED FUNCTIONS 2 A BIT OF TOPOLOGY IN RN 3 LIMITS AND CONTINUITY 3 THE DERIVATIVE 1 PARTIAL DERIVATIVES AND DIRECTIONAL DERIVATIVES 2 DIFFERENTIABILITY 3 DIFFERENTIATION RULES 4 THE GRADIENT 5 CURVES 6 HIGHER-ORDER PARTIAL DERIVATIVES 4 IMPLICIT AND EXPLICIT SOLUTIONS OF LINEAR SYSTEMS 1 GAUSSIAN ELIMINATION ANDTHE THEORY OF LINEAR SYSTEMS 2 ELEMENTARY MATRICES ANDCALCULATING INVERSE MATRICES 3 LINEAR INDEPENDENCE, BASIS, AND DIMENSION 4 THE FOUR FUNDAMENTAL SUBSPACES 5 EXTREMUM PROBLEMS 1 COMPACTNESS AND THE MAXIMUM VALUE THEOREM 2 MAXIMUM/MINIMUM PROBLEMS 3 QUADRATIC FORMS AND THE SECOND DERIVATIVE TEST 4 LAGRANGE MULTIPLIERS 5 PROJECTIONS, LEAST SQUARES,AND INNER PRODUCT SPACES 6 SOLVING NONLINEAR PROBLEMS 1 THE CONTRACTION MAPPING PRINCIPLE 2 THE INVERSE AND IMPLICIT FUNCTION THEOREMS 3 MANIFOLDS REVISITED 7 INTEGRATION 1 MULTIPLE INTEGRALS 2 ITERATED INTEGRALS AND FUBINI’S THEOREM 3 POLAR, CYLINDRICAL, AND SPHERICAL COORDINATES 4 PHYSICAL APPLICATIONS 5 DETERMINANTS AND n -DIMENSIONAL VOLUME 6 CHANGE OF VARIABLES THEOREM 8 DIFFERENTIAL FORMSAND INTEGRATION ON MANIFOLDS 1 MOTIVATION 2 DIFFERENTIAL FORMS 3 LINE INTEGRALS AND GREEN’S THEOREM 4 SURFACE INTEGRALS AND FLUX 5 STOKES’S THEOREM 6 APPLICATIONS TO PHYSICS 7 APPLICATIONS TO TOPOLOGY 9 EIGENVALUES,EIGENVECTORS, AND APPLICATIONS 1 LINEAR TRANSFORMATIONS AND CHANGE OF BASIS 2 EIGENVALUES, EIGENVECTORS, AND DIAGONALIZABILITY 3 DIFFERENCE EQUATIONS AND ORDINARY DIFFERENTIAL EQUATIONS 4 THE SPECTRAL THEOREM GLOSSARY FURTHER READING ANSWERS TO SELECTED EXERCISES INDEX