多元微积分教程

1Vectors and Functions1.1 PreliminariesAlgebraic OperationsOrder PropertiesIntervals， Disks， and Bounded SetsLine Segments and Paths1.2 Functions and Their Geometric PropertiesBasic NotionsBounded FunctionsMonotonicity and BimonotonicityFunctions of Bounded VariationFunctions of Bounded BivariationConvexity and ConcavityLocal Extrema and Saddle PointsIntermediate Value Property1.3 Cylindricaj and Spherical CoordinatesCylindrical CoordinatesSpherical CoordinatesNotes and CommentsExercises2 Sequences， Continuity， and Limits2.1 Sequences in R2， Subsequences and Cauchy SequencesClosure， Boundary， and Interior2.2 ContinuityComposition of Continuous FunctionsPiecing Continuous Functions on Overlapping SubsetsCharacterizations of ContinuityContinuity and BoundednessContinuity and MonotonicityContinuity， Bounded Variation， and Bounded BivariationContinuity and ConvexityContinuity and Intermediate Value PropertyUniform ContinuityImplicit Function Theorem2.3 LimitsLimits and ContinuityLimit from a QuadrantApproaching InfinityNotes and CommentsExercises3 Partial and Total Differentiation3.1 Partial and Directional DerivativesPartial DerivativesDirectional DerivativesHigher-Order Partial DerivativesHigher-Order Directional Derivatives3.2 DifferentiabilityDifferentiability and Directional DerivativesImplicit Differentiation3.3 Taylor's Theorem and Chain RuleBivariate Taylor TheoremChain Rule3.4 Monotonicity and ConvexityMonotonicity and First PartialsBimonotonicity and Mixed PartialsBounded Variation and Boundedness of First PartialsBounded Bivariation and Boundedness of Mixed PartialsConvexity and Monotonicity of GradientConvexity and Nonnegativity of Hessian3.5 Functions of Three Variables. ，Extensions and AnaloguesTangent Planes and Normal Lines to SurfacesConvexity and Ternary Quadratic FormsNotes and CommentsExercises4 Applications of Partial Differentiation4.1 Absolute ExtremaBoundary Points and Critical Points4.2 Constrained ExtremaLagrange Multiplier MethodCase of Three Variables4.3 Local Extrema and Saddle Points……5 Multiple Integration6 Applications and Approximations of Multiple Integrals7Double Series and Improper Double IntegralsReferencesList of Symbols and AbbreviationsIndex

Calculus of real-valued functions of several real variables， also known as multivariable calculus， is a rich and fascinating subject. On the one hand， it seeks to extend eminently useful and immensely successful notions in one-variable calculus such as limit， continLuty， derivative， and integral to "higher dimensions." On the other hand， the fact that there is much more room to move about in the 'n space Rn than on the real line R brings to the fore deeper geometric and topological notions that play a significant role in the study of functions of two or more variables.