A course in multivariable calculus and analysis /

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Bibliographic Details
Author / Creator: Ghorpade, Sudhir, 1963-
Other Authors / Creators:Limaye, Balmohan Vishnu.
Format: Book
Language:English
Imprint: New York : Springer, [2010]
Series:Undergraduate texts in mathematics.
Subjects:
Table of Contents:
  • 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 R 2
  • 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
  • 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 R 2
  • 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 R 3
  • 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 R 2
  • Regions between Polar Curves
  • Volume of a Bounded Subset of R 3
  • 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 R 2 and in R 3
  • 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
  • Functions on Unbounded Subsets
  • Concept of Area of an Unbounded Subset of R 2
  • Unbounded Functions on Bounded Subsets
  • Notes and Comments
  • Exercises
  • References
  • List of Symbols and Abbreviations
  • Index