Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds

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