Schaum's Outline of Advanced Calculus

Description

Content: Intro -- Copyright -- PREFACE -- CONTENTS -- CHAPTER 1 Numbers -- SETS -- REAL NUMBERS -- DECIMAL REPRESENTATION OF REAL NUMBERS -- GEOMETRIC REPRESENTATION OF REAL NUMBERS -- OPERATIONS WITH REAL NUMBERS -- INEQUALITIES -- ABSOLUTE VALUE OF REAL NUMBERS -- EXPONENTS AND ROOTS -- LOGARITHMS -- AXIOMATIC FOUNDATIONS OF THE REAL NUMBER SYSTEM -- POINT SETS, INTERVALS -- COUNTABILITY -- NEIGHBORHOODS -- LIMIT POINTS -- BOUNDS -- BOLZANO -WEIERSTRASS THEOREM -- ALGEBRAIC AND TRANSCENDENTAL NUMBERS -- THE COMPLEX NUMBER SYSTEM -- POLAR FORM OF COMPLEX NUMBERS -- MATHEMATICAL INDUCTION -- CHAPTER 2 Sequences -- DEFINITION OF A SEQUENCE -- LIMIT OF A SEQUENCE -- THEOREMS ON LIMITS OF SEQUENCES -- INFINITY -- BOUNDED, MONOTONIC SEQUENCES -- LEAST UPPER BOUND AND GREATEST LOWER BOUND OF A SEQUENCE -- LIMIT SUPERIOR, LIMIT INFERIOR -- NESTED INTERVALS -- CAUCHY'S CONVERGENCE CRITERION -- INFINITE SERIES -- CHAPTER 3 Functions, Limits, and Continuity -- FUNCTIONS -- GRAPH OF A FUNCTION -- BOUNDED FUNCTIONS -- MONOTONIC FUNCTIONS -- INVERSE FUNCTIONS. PRINCIPAL VALUES -- MAXIMA AND MINIMA -- TYPES OF FUNCTIONS -- TRANSCENDENTAL FUNCTIONS -- LIMITS OF FUNCTIONS -- RIGHT-AND LEFT-HAND LIMITS -- THEOREMS ON LIMITS -- INFINITY -- SPECIAL LIMITS -- CONTINUITY -- RIGHT-AND LEFT-HAND CONTINUITY -- CONTINUITY IN AN INTERVAL -- THEOREMS ON CONTINUITY -- PIECEWISE CONTINUITY -- UNIFORM CONTINUITY -- CHAPTER 4 Derivatives -- THE CONCEPT AND DEFINITION OF A DERIVATIVE -- RIGHT-AND LEFT-HAND DERIVATIVES -- DIFFERENTIABILITY IN AN INTERVAL -- PIECEWISE DIFFERENTIABILITY -- DIFFERENTIALS -- THE DIFFERENTIATION OF COMPOSITE FUNCTIONS -- IMPLICIT DIFFERENTIATION -- RULES FOR DIFFERENTIATION -- DERIVATIVES OF ELEMENTARY FUNCTIONS -- HIGHER ORDER DERIVATIVES -- MEAN VALUE THEOREMS -- L'HOSPITAL'S RULES -- APPLICATIONS -- CHAPTER 5 Integrals. INTRODUCTION OF THE DEFINITE INTEGRAL -- MEASURE ZERO -- PROPERTIES OF DEFINITE INTEGRALS -- MEAN VALUE THEOREMS FOR INTEGRALS -- CONNECTING INTEGRAL AND DIFFERENTIAL CALCULUS -- THE FUNDAMENTAL THEOREM OF THE CALCULUS -- GENERALIZATION OF THE LIMITS OF INTEGRATION -- CHANGE OF VARIABLE OF INTEGRATION -- INTEGRALS OF ELEMENTARY FUNCTIONS -- SPECIAL METHODS OF INTEGRATION -- IMPROPER INTEGRALS -- NUMERICAL METHODS FOR EVALUATING DEFINITE INTEGRALS -- APPLICATIONS -- ARC LENGTH -- AREA -- VOLUMES OF REVOLUTION -- CHAPTER 6 Partial Derivatives -- FUNCTIONS OF TWO OR MORE VARIABLES -- THREE-DIMENSIONAL RECTANGULAR COORDINATE SYSTEMS -- NEIGHBORHOODS -- REGIONS -- LIMITS -- ITERATED LIMITS -- CONTINUITY -- UNIFORM CONTINUITY -- PARTIAL DERIVATIVES -- HIGHER ORDER PARTIAL DERIVATIVES -- DIFFERENTIALS -- THEOREMS ON DIFFERENTIALS -- DIFFERENTIATION OF COMPOSITE FUNCTIONS -- EULER'S THEOREM ON HOMOGENEOUS FUNCTIONS -- IMPLICIT FUNCTIONS -- JACOBIANS -- PARTIAL DERIVATIVES USING JACOBIANS -- THEOREMS ON JACOBIANS -- TRANSFORMATIONS -- CURVILINEAR COORDINATES -- MEAN VALUE THEOREM -- CHAPTER 7 Vectors -- VECTORS -- GEOMETRIC PROPERTIES -- ALGEBRAIC PROPERTIES OF VECTORS -- LINEAR INDEPENDENCE AND LINEAR DEPENDENCE OF A SET OF VECTORS -- UNIT VECTORS -- RECTANGULAR (ORTHOGONAL) UNIT VECTORS -- COMPONENTS OF A VECTOR -- DOT OR SCALAR PRODUCT -- CROSS OR VECTOR PRODUCT -- TRIPLE PRODUCTS -- AXIOMATIC APPROACH TO VECTOR ANALYSIS -- VECTOR FUNCTIONS -- LIMITS, CONTINUITY, AND DERIVATIVES OF VECTOR FUNCTIONS -- GEOMETRIC INTERPRETATION OF A VECTOR DERIVATIVE -- GRADIENT, DIVERGENCE, AND CURL -- FORMULAS INVOLVING v -- VECTOR INTERPRETATION OF JACOBIANS, ORTHOGONAL CURVILINEAR COORDINATES -- GRADIENT DIVERGENCE, CURL, AND LAPLACIAN IN ORTHOGONAL CURVILINEAR COORDINATES -- SPECIAL CURVILINEAR COORDINATES -- CHAPTER 8 Applications of Partial Derivatives. APPLICATIONS TO GEOMETRY -- DIRECTIONAL DERIVATIVES -- DIFFERENTIATION UNDER THE INTEGRAL SIGN -- INTEGRATION UNDER THE INTEGRAL SIGN -- MAXIMA AND MINIMA -- METHOD OF LAGRANGE MULTIPLIERS FOR MAXIMA AND MINIMA -- APPLICATIONS TO ERRORS -- CHAPTER 9 Multiple Integrals -- DOUBLE INTEGRALS -- ITERATED INTEGRALS -- TRIPLE INTEGRALS -- TRANSFORMATIONS OF MULTIPLE INTEGRALS -- THE DIFFERENTIAL ELEMENT OF AREA IN POLAR COORDINATES, DIFFERENTIAL ELEMENTS OF AREA IN CYLINDRAL AND SPHERICAL COORDINATES -- CHAPTER 10 Line Integrals, Surface Integrals, and Integral Theorems -- LINE INTEGRALS -- EVALUATION OF LINE INTEGRALS FOR PLANE CURVES -- PROPERTIES OF LINE INTEGRALS EXPRESSED FOR PLANE CURVES -- SIMPLE CLOSED CURVES, SIMPLY AND MULTIPLY CONNECTED REGIONS -- GREEN'S THEOREM IN THE PLANE -- CONDITIONS FOR A LINE INTEGRAL TO BE INDEPENDENT OF THE PATH -- SURFACE INTEGRALS -- THE DIVERGENCE THEOREM -- STOKES' THEOREM -- CHAPTER 11 Infinite Series -- DEFINITIONS OF INFINITE SERIES AND THEIR CONVERGENCE AND DIVERGENCE -- FUNDAMENTAL FACTS CONCERNING INFINITE SERIES -- SPECIAL SERIES -- TESTS FOR CONVERGENCE AND DIVERGENCE OF SERIES OF CONSTANTS -- THEOREMS ON ABSOLUTELY CONVERGENT SERIES -- INFINITE SEQUENCES AND SERIES OF FUNCTIONS, UNIFORM CONVERGENCE -- SPECIAL TESTS FOR UNIFORM CONVERGENCE OF SERIES -- THEOREMS ON UNIFORMLY CONVERGENT SERIES -- POWER SERIES -- THEOREMS ON POWER SERIES -- OPERATIONS WITH POWER SERIES -- EXPANSION OF FUNCTIONS IN POWER SERIES -- TAYLOR'S THEOREM -- SOME IMPORTANT POWER SERIES -- SPECIAL TOPICS -- TAYLOR'S THEOREM (FOR TWO VARIABLES) -- CHAPTER 12 Improper Integrals -- DEFINITION OF AN IMPROPER INTEGRAL -- IMPROPER INTEGRALS OF THE FIRST KIND (Unbounded Intervals) -- CONVERGENCE OR DIVERGENCE OF IMPROPER INTEGRALS OF THE FIRST KIND -- SPECIAL IMPROPER INTEGRALS OF THE FIRST KIND. CONVERGENCE TESTS FOR IMPROPER INTEGRALS OF THE FIRST KIND -- IMPROPER INTEGRALS OF THE SECOND KIND -- CAUCHY PRINCIPAL VALUE -- SPECIAL IMPROPER INTEGRALS OF THE SECOND KIND -- CONVERGENCE TESTS FOR IMPROPER INTEGRALS OF THE SECOND KIND -- IMPROPER INTEGRALS OF THE THIRD KIND -- IMPROPER INTEGRALS CONTAINING A PARAMETER, UNIFORM CONVERGENCE -- SPECIAL TESTS FOR UNIFORM CONVERGENCE OF INTEGRALS -- THEOREMS ON UNIFORMLY CONVERGENT INTEGRALS -- EVALUATION OF DEFINITE INTEGRALS -- LAPLACE TRANSFORMS -- LINEARITY -- CONVERGENCE -- APPLICATION -- IMPROPER MULTIPLE INTEGRALS -- CHAPTER 13 Fourier Series -- PERIODIC FUNCTIONS -- FOURIER SERIES -- ORTHOGONALITY CONDITIONS FOR THE SINE AND COSINE FUNCTIONS -- DIRICHLET CONDITIONS -- ODD AND EVEN FUNCTIONS -- HALF RANGE FOURIER SINE OR COSINE SERIES -- PARSEVAL'S IDENTITY -- DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES -- COMPLEX NOTATION FOR FOURIER SERIES -- BOUNDARY-VALUE PROBLEMS -- ORTHOGONAL FUNCTIONS -- CHAPTER 14 Fourier Integrals -- THE FOURIER INTEGRAL -- EQUIVALENT FORMS OF FOURIER'S INTEGRAL THEOREM -- FOURIER TRANSFORMS -- CHAPTER 15 Gamma and Beta Functions -- THE GAMMA FUNCTION -- TABLES OF VALUES AND GRAPH OF THE GAMMA FUNCTION -- THE BETA FUNCTION -- DIRICHLET INTEGRALS -- CHAPTER 16 Functions of a Complex Variable -- FUNCTIONS -- LIMITS AND CONTINUITY -- DERIVATIVES -- CAUCHY-RIEMANN EQUATIONS -- INTEGRALS -- CAUCHY'S THEOREM -- CAUCHY'S INTEGRAL FORMULAS -- TAYLOR'S SERIES -- SINGULAR POINTS -- POLES -- LAURENT'S SERIES -- BRANCHES AND BRANCH POINTS -- RESIDUES -- RESIDUE THEOREM -- EVALUATION OF DEFINITE INTEGRALS -- INDEX.