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A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics)by Sudhir R. Ghorpade
Synopses & ReviewsPublisher Comments:This selfcontained textbook gives a thorough exposition of multivariable calculus. It can be viewed as a sequel to the onevariable calculus text, A Course in Calculus and Real Analysis, published in the same series. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in onevariable calculus. For example, when the general definition of the volume of a solid is given using triple integrals, the authors explain why the shell and washer methods of onevariable calculus for computing the volume of a solid of revolution must give the same answer. Further, the book includes genuine analogues of basic results in onevariable calculus, such as the mean value theorem and the fundamental theorem of calculus. This book is distinguished from others on the subject: it examines topics not typically covered, such as monotonicity, bimonotonicity, and convexity, together with their relation to partial differentiation, cubature rules for approximate evaluation of double integrals, and conditional as well as unconditional convergence of double series and improper double integrals. Moreover, the emphasis is on a geometric approach to such basic notions as local extremum and saddle point. Each chapter contains detailed proofs of relevant results, along with numerous examples and a wide collection of exercises of varying degrees of difficulty, making the book useful to undergraduate and graduate students alike. There is also an informative section of "Notes and Comments indicating some novel features of the treatment of topics in that chapter as well as references to relevant literature. The only prerequisite for this text is a course in onevariable calculus.
Table of Contents1 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 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. 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 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. Iterated Series. Telescoping Double Series. Double Series with Nonnegative Terms. Absolute Convergence and Conditional 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 Related Double Integrals. Functions on Unbounded Subsets. Area of an Unbounded Set in R2. Unbounded Functions on Bounded Subsets. Exercises. References. Index
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