A TEXTBOOK OF VECTOR ALGEBRA
CONTENTS CHAPTER INTRODUCTION READING MATHEMATICS QUANTI?ERS AND NEGATION SET THEORY FUNCTIONS REAL NUMBERS IN?NITE SETS COMPLEX NUMBERS CHAPTER INTRODUCING THE ACTORS: POINTS AND VECTORS INTRODUCING THE ACTORS: MATRICES MATRIX MULTIPLICATION AS A LINEAR TRANSFORMATION THE GEOMETRY OF RN LIMITS AND CONTINUITY FIVE BIG THEOREMS DERIVATIVES IN SEVERAL VARIABLES AS LINEAR TRANSFORMATIONS RULES FOR COMPUTING DERIVATIVES THE MEAN VALUE THEOREM AND CRITERIA FOR DI?ERENTIABILITY 1 10 REVIEW EXERCISES FOR CHAPTER THE MAIN ALGORITHM: ROW REDUCTION SOLVING EQUATIONS WITH ROW REDUCTION MATRIX INVERSES AND ELEMENTARY MATRICES LINEAR COMBINATIONS, SPAN, AND LINEAR INDEPENDENCE KERNELS, IMAGES, AND THE DIMENSION FORMULA ABSTRACT VECTOR SPACES EIGENVECTORS AND EIGENVALUES NEWTON’S METHOD SUPERCONVERGENCE 2 10 THE INVERSE AND IMPLICIT FUNCTION THEOREMS 2 11 REVIEW EXERCISES FOR CHAPTER MANIFOLDS TANGENT SPACES TAYLOR POLYNOMIALS IN SEVERAL VARIABLES RULES FOR COMPUTING TAYLOR POLYNOMIALS QUADRATIC FORMS CLASSIFYING CRITICAL POINTS OF FUNCTIONS CONSTRAINED CRITICAL POINTS AND LAGRANGE MULTIPLIERS PROBABILITY AND THE SINGULAR VALUE DECOMPOSITION GEOMETRY OF CURVES AND SURFACES 3 10 REVIEW EXERCISES FOR CHAPTER DE?NING THE INTEGRAL PROBABILITY AND CENTERS OF GRAVITY WHAT FUNCTIONS CAN BE INTEGRATED? MEASURE ZERO FUBINI’S THEOREM AND ITERATED INTEGRALS NUMERICAL METHODS OF INTEGRATION OTHER PAVINGS DETERMINANTS VOLUMES AND DETERMINANTS 4 10 THE CHANGE OF VARIABLES FORMULA 4 11 LEBESGUE INTEGRALS 4 12 REVIEW EXERCISES FOR CHAPTER PARALLELOGRAMS AND THEIR VOLUMES PARAMETRIZATIONS COMPUTING VOLUMES OF MANIFOLDS INTEGRATION AND CURVATURE FRACTALS AND FRACTIONAL DIMENSION REVIEW EXERCISES FOR CHAPTER FORMS ON RN INTEGRATING FORM ?ELDS OVER PARAMETRIZED DOMAINS ORIENTATION OF MANIFOLDS INTEGRATING FORMS OVER ORIENTED MANIFOLDS FORMS IN THE LANGUAGE OF VECTOR CALCULUS BOUNDARY ORIENTATION THE EXTERIOR DERIVATIVE GRAD, CURL, DIV, AND ALL THAT THE PULLBACK 6 10 THE GENERALIZED STOKES’S THEOREM 6 11 THE INTEGRAL THEOREMS OF VECTOR CALCULUS 6 12 ELECTROMAGNETISM 6 13 POTENTIALS 6 14 REVIEW EXERCISES FOR CHAPTER