A Course in Multivariable Calculus and Analysis

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.- Basic Examples.- 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.- Subsequences and Cauchy Sequences.- Closure, Boundary, and Interior.- 2.2 Continuity.- Composition of Continuous Functions.- Piecing Continuous Functions on Overlapping Subsets.- Characterizations of Continuity.- Continuity and Boundedness.- Continuity and Monotonicity.- 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.- Limits along 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 Linesto 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.- A Basic Inequality and a 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 Set in R2.- Domain Additivity over Bounded Sets.- 5.3 Change of Variables.- Translation Invariance and Area of a Parallelogram.- Case of Affine Transformations.- General Case.- Polar Coordinates.- 5.4 Triple Integrals.- Triple Integrals over Bounded Sets.- Sets of Three Dimensional Content Zero.- Concept of Volume of a Bounded Set in 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 Set in R2.- Regions between Polar Curves.- Volume of a Bounded Set in 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