本书是一本全面介绍了多元微积分知识的教程。重点强调相关的基本概念和与多元微积分相关的一元微积分部分。书中总结了单变量微积分类似的基本结果，如均值定理和微积分基础定理。这本书和该科目的其他书的最大不同点是：包罗了经典中不包含的单调性、双单调性、凸性以及它们和偏微分方程的关系、二重积分近似值的体积规则、二重级数和反常二重级数的条件和无条件收敛。每章都包括相关结果的详细证明、大量的例子和不同难度的练习，使得这本书对本科生和研究生同等重要。读者对象：数学专业的本科生、研究生和理工科的数学基础课。
1 Vectors and Functions
1.1 Preliminaries
Algebraic Operations
Order Properties
Intervals, Disks, and Bounded Sets
Line Segments and Paths
1.2 Functions and Their Geometric Properties
Basic Notions
Bounded Functions
Monotonicity and Bimonotonicity
Functions of Bounded Variation
Functions of Bounded Bivariation
Convexity and Concavity
Local Extrema and Saddle Points
Intermediate Value Property
1.3 Cylindrical and Spherical Coordinates
Cylindrical Coordinates
Spherical Coordinates
Notes and Comments
Exercises
2 Sequences, Continuity, and Limits
2.1 Sequences in R2
SubsequenCec Slosure' Boundara y,nd Cauchy Sequencea Snd Interior
2.2 Continuity
Composition of Continuous Functions
Piecing Continuous Functions on Overlapping Subsets
Characterizations of Continuity
Continuity and Boundedness
Continuity and Monotouicity
Continuity, Bounded Variation, and Bounded Bivariation
Continuity and Convexity
Continuity and Intermediate Value Property
Uniform Continuity
Implicit Function Theorem
2.3 Limits
Limits and Continuity
Limit from a Quadrant
Approaching Infinity
Notes and Comments
Exercises
3 Partial and Total Differentiation
3.1 Partial and Directional Derivatives
Partial Derivatives
Directional Derivatives
Higher-Order Partial Derivatives
Higher-Order Directional Derivatives
3.2 Differentiability
Differentiability and Directional Derivatives
Implicit Differentiation
3.3 Taylor's Theorem and Chain Rule
Bivariate Taylor Theorem
Chain Rule
3.4 Monotonicity and Convexity
Monotonicity and First Partials
Bimonotonicity and Mixed Partials
Bounded Variation and Boundedness of First Partials
Bounded Bivariation and Boundedness of Mixed Partials
Convexity and Monotonicity of Gradient
Convexity and Nonnegativity of Hessian
3.5 Functions of Three Variables
Extensions and Analogues
Tangent Planes and Normal Lines to Surfaces
Convexity and Ternary Quadratic Forms
Notes and Comments
Exercises
4 Applications of Partial Differentiation
4.1 Absolute Extrema
Boundary Points and Critical Points
4.2 Constrained Extrema
Lagrange Multiplier Method
Case of Three Variables
4.3 Local Extrema and Saddle Points
Discriminant Test
4.4 Linear and Quadratic Approximations
Linear Approximation
Quadratic Approximation
Notes and Comments
Exercises
5 Multiple Integration
5.1 Double Integrals on Rectangles
Basic Inequality and Criterion for Integrability
Domain Additivity on Rectangles
Integrability of Monotonic and Continuous Functions
Algebraic and Order Properties
A Version of the Fundamental Theorem of Calculus
Fubini's Theorem on Rectangles
Riemann Double Sums
5.2 Double Integrals over Bounded Sets
Fubini's Theorem over Elementary Regions
Sets of Content Zero
Concept of Area of a Bounded Subset of R2
Domain Additivity over Bounded Sets
5.3 Change of Variables
Translation Invariance and Area of a Parallelogram
Case of Affine Transformations
General Case
5.4 Triple Integrals
Triple Integrals over Bounded Sets
Sets of Three-Dimensional Content Zero
Concept of Volume of a Bounded Subset of R3
Change of Variables in Triple Integrals
Notes and Comments
Exercises
6 Applications and Approximations of Multiple Integrals
6.1 Area and Volume
Area of a Bounded Subset of R2
Regions between Polar Curves
Volume of a Bounded Subset of R3
Solids between Cylindrical or Spherical Surfaces
Slicing by Planes and the Washer Method
Slivering by Cylinders and the Shell Method
6.2 Surface Area
Parallelograms in R2 and in R3
Area of a Smooth Surface
Surfaces of Revolution
6.3 Centroids of Surfaces and Solids
Averages and Weighted Averages
Centroids of Planar Regions
Centroids of Surfaces
Centroids of Solids
Centroids of Solids of Revolution
6.4 Cubature Rules
Product Rules on Rectangles
Product Rules over Elementary Regions
Triangular Prism Rules
Notes and Comments
Exercises
7 Double Series and Improper Double Integrals
7.1 Double Sequences
Monotonicity and Bimonotonicity
7.2 Convergence of Double Series
Telescoping Double Series
Double Series with Nonnegative Terms
Absolute Convergence and Conditional Convergence
Unconditional Convergence
7.3 Convergence Tests for Double Series
Tests for Absolute Convergence
Tests for Conditional Convergence
7.4 Double Power Series
Taylor Double Series and Taylor Series
7.5 Convergence of Improper Double Integrals
Improper Double Integrals of Mixed Partials
Improper Double Integrals of Nonnegative Functions
Absolute Convergence and Conditional Convergence
7.6 Convergence Tests for Improper Double Integrals
Tests for Absolute Convergence
Tests for Conditional Convergence
7.7 Unconditional Convergence of Improper Double Integrals
on Unctions on Unbounded Subsets
Concept of Area of an Unbounded Subset of R2
Unbounded Functions on Bounded Subsets
Notes and Comments
Exercises
References
List of Symbols and Abbreviations
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