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 FUBINIβS 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 GREENβS THEOREM
4 SURFACE INTEGRALS AND FLUX
5 STOKESβS 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