# A Course in Multivariable Calculus and Analysis

A Course in Multivariable Calculus and Analysis This self-contained textbook gives a thorough exposition of multivariable calculus. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in one-variable calculus. Further, the book includes genuine analogues of basic results in one-variable 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. 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. ; This self-contained textbook gives a thorough exposition of multivariable calculus. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in one-variable calculus. 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. ; Calculus of real-valued functions of several real variables, also known as m- tivariable 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, continuity, derivative, and integral to “higher dim- sions. ” On the other hand, the fact that there is much more room to move n about in the n-space R than on the real line R brings to the fore deeper geometric and topological notions that play a signi?cant role in the study of functions of two or more variables. Courses in multivariable calculus at an undergraduate level and even at an advanced level are often faced with the unenviable task of conveying the multifarious and multifaceted aspects of multivariable calculus to a student in the span of just about a semester or two. Ambitious courses and teachers would try to give some idea of the general Stokes’s theorem for di?erential forms on manifolds as a grand generalization of the fundamental theorem of calculus, and prove the change of variables formula in all its glory. They would also try to do justice to important results such as the implicit function theorem, which really have no counterpart in one-variable calculus. Most courses would require the student to develop a passing acquaintance with the theorems of Green, Gauss, and Stokes, never mind the tricky questions about orientability, simple connectedness, etc.; 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; From the reviews: “There is no doubt that one of the key mathematical courses, perhaps the most important and fundamental one for undergraduates in various branches of science and engineering, is calculus. … This is essentially a textbook suitable for a one-semester course in multivariable calculus or analysis for undergraduates in mathematics. … it contains some material that would be very useful for engineers. … I recommend this book … for undergraduate students in mathematics and professors teaching courses in multivariable calculus.” (Mehdi Hassani, The Mathematical Association of America, June, 2010) “This book was written as a textbook for a second course in calculus … . The authors differentiate this book from many similar works in terms of the continuity of approach between one-variable calculus and multivariable calculus, as well as the addition of several unique topics. The book is self-contained … . Summing Up: Recommended. Lower- and upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 47 (11), July, 2010) “This text is a fairly thorough treatment of real multivariable calculus which aims to develop wherever possible notions and results analogous to those in one-variable calculus. … Each chapter concludes with a section of notes and comments, and an extensive set of exercises.” (Gerald A. Heuer, Zentralblatt MATH, Vol. 1186, 2010) ; This self-contained textbook gives a thorough exposition of multivariable calculus. It can be viewed as a sequel to the one-variable 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 one-variable 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 one-variable 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 one-variable 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 one-variable calculus. ; Self-contained Neatly ties up multivariable calculus with its relics in one variable calculus Caters to theoretical as well as practical aspects of multivariable calculus Contains extensive material on topics not typically covered in multivariable calculus textbooks, such as: monotonicity and bimonotonicity of functions of two variables and their relationship with partial differentiation; higher order directional derivatives and their use in Taylor's Theorem; US http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Course in Multivariable Calculus and Analysis

486 pages

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Publisher
Springer New York
Copyright
Copyright ï¿½ Springer Basel AG
DOI
10.1007/978-1-4419-1621-1
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### Abstract

This self-contained textbook gives a thorough exposition of multivariable calculus. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in one-variable calculus. Further, the book includes genuine analogues of basic results in one-variable 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. 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. ; This self-contained textbook gives a thorough exposition of multivariable calculus. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in one-variable calculus. 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. ; Calculus of real-valued functions of several real variables, also known as m- tivariable 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, continuity, derivative, and integral to “higher dim- sions. ” On the other hand, the fact that there is much more room to move n about in the n-space R than on the real line R brings to the fore deeper geometric and topological notions that play a signi?cant role in the study of functions of two or more variables. Courses in multivariable calculus at an undergraduate level and even at an advanced level are often faced with the unenviable task of conveying the multifarious and multifaceted aspects of multivariable calculus to a student in the span of just about a semester or two. Ambitious courses and teachers would try to give some idea of the general Stokes’s theorem for di?erential forms on manifolds as a grand generalization of the fundamental theorem of calculus, and prove the change of variables formula in all its glory. They would also try to do justice to important results such as the implicit function theorem, which really have no counterpart in one-variable calculus. Most courses would require the student to develop a passing acquaintance with the theorems of Green, Gauss, and Stokes, never mind the tricky questions about orientability, simple connectedness, etc.; 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; From the reviews: “There is no doubt that one of the key mathematical courses, perhaps the most important and fundamental one for undergraduates in various branches of science and engineering, is calculus. … This is essentially a textbook suitable for a one-semester course in multivariable calculus or analysis for undergraduates in mathematics. … it contains some material that would be very useful for engineers. … I recommend this book … for undergraduate students in mathematics and professors teaching courses in multivariable calculus.” (Mehdi Hassani, The Mathematical Association of America, June, 2010) “This book was written as a textbook for a second course in calculus … . The authors differentiate this book from many similar works in terms of the continuity of approach between one-variable calculus and multivariable calculus, as well as the addition of several unique topics. The book is self-contained … . Summing Up: Recommended. Lower- and upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 47 (11), July, 2010) “This text is a fairly thorough treatment of real multivariable calculus which aims to develop wherever possible notions and results analogous to those in one-variable calculus. … Each chapter concludes with a section of notes and comments, and an extensive set of exercises.” (Gerald A. Heuer, Zentralblatt MATH, Vol. 1186, 2010) ; This self-contained textbook gives a thorough exposition of multivariable calculus. It can be viewed as a sequel to the one-variable 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 one-variable 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 one-variable 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 one-variable 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 one-variable calculus. ; Self-contained Neatly ties up multivariable calculus with its relics in one variable calculus Caters to theoretical as well as practical aspects of multivariable calculus Contains extensive material on topics not typically covered in multivariable calculus textbooks, such as: monotonicity and bimonotonicity of functions of two variables and their relationship with partial differentiation; higher order directional derivatives and their use in Taylor's Theorem; US

Published: Mar 20, 2010