Wind, Sand and Stars

"This textbook offers an extensive list of completely solved problems in mathematical analysis. This second of three volumes covers definite, improper and multidimensional integrals, functions of several variables, differential equations, and more. The series contains the material corresponding to the first three or four semesters of a course in Mathematical Analysis. Based on the authors years of teaching experience, this work stands out by providing detailed solutions (often several pages long) to the problems. The basic premise of the book is that no topic should be left unexplained, and no question that could realistically arise while studying the solutions should remain unanswered. The style and format are straightforward and accessible. In addition, each chapter includes exercises for students to work on independently. Answers are provided to all problems, allowing students to check their work. Though chiefly intended for early undergraduate students of Mathematics, Physics and Engineering, the book will also appeal to students from other areas with an interest in Mathematical Analysis, either as supplementary reading or for independent study"--Provided by publisher Preface Contents Definitions and Notation 1 Exploring the Riemann and Definite Integral 1.1 Examining the Integrability of Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 1.2 Finding Riemann Integrals by Definition Problem 1 Solution Problem 2 Solution Problem 3 Solution 1.3 Finding Limits of Certain Sequences Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 1.4 Calculating Various Interesting Definite Integrals Problem 1 Solution Problem 2 Solution Integral (a) Integral (b) Integral (c) Problem 3 Solution Problem 4 Solution Problem 5 Solution Problem 6 Solution 1.5 Explaining Several Apparent Paradoxes Problem 1 Solution Problem 2 Solution Problem 3 Solution 1.6 Using the (Second) Mean Value Theorem Problem 1 Solution Problem 2 Solution Problem 3 Solution 1.7 Exercises for Independent Work 2 Examining Improper Integrals 2.1 Investigating the Convergence of Integrals by Definition Problem 1 Solution Problem 2 Solution 2.2 Using Different Criteria Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 2.3 Using Integral Test for Convergence of Series Problem 1 Solution Problem 2 Solution 2.4 Exercises for Independent Work 3 Applying One-Dimensional Integrals to Geometry and Physics 3.1 Finding Lengths of Curves Problem 1 Solution Problem 2 Solution Problem 3 Solution 3.2 Calculating Areas of Surfaces Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 3.3 Finding Volumes and Surface Areas of Solids of Revolution Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 3.4 Finding Various Physical Quantities Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 3.5 Exercises for Independent Work 4 Dealing with Functions of Several Variables 4.1 Finding Images and Preimages of Sets Problem 1 Solution Problem 2 Solution 4.2 Examining Limits and Continuity of Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 4.3 Exercises for Independent Work 5 Investigating Derivatives of Multivariable Functions 5.1 Calculating Partial and Directional Derivatives Problem 1 Solution Problem 2 Solution Problem 3 Solution 5.2 Examining Differentiability of Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 5.3 Exercises for Independent Work 6 Examining Higher Derivatives, Differential Expressions, and Taylor's Formula 6.1 Verifying the Existence of the Second Derivatives Problem 1 Solution Problem 2 Solution 6.2 Transforming Differential Expressions and Operators Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 6.3 Expanding Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution 6.4 Exercises for Independent Work 7 Examining Extremes and Other Important Points 7.1 Looking for Global Maxima and Minima of Functions on Compact Sets Problem 1 Solution Problem 2 Solution Problem 3 Solution 7.2 Examining Local Extremes and Saddle Points of Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution 7.3 Exercises for Independent Work 8 Examining Implicit and Inverse Functions 8.1 Investigating the Existence and Extremes of Implicit Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 8.2 Finding Derivatives of Inverse Functions Problem 1 Solution Problem 2 Solution 8.3 Exercises for Independent Work 9 Solving Differential Equations of the First Order 9.1 Finding Solutions of Separable Equations Problem 1 Solution Problem 2 Solution Problem 3 Solution 9.2 Solving Homogeneous Equations Problem 1 Solution Problem 2 Solution Problem 3 Solution 9.3 Solving Several Specific Equations Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 9.4 Solving Exact Equations Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 9.5 Exercises for Independent Work 10 Solving Differential Equations of Higher Orders 10.1 Solving Linear Equations with Constant Coefficients Problem 1 Solution Problem 2 Solution Problem 3 Solution 10.2 Using Various Specific Methods Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 10.3 Exercises for Independent Work 11 Solving Systems of First-Order Differential Equations 11.1 Using the Method of Elimination of Variables Problem 1 Solution Problem 2 Solution Problem 3 Solution 11.2 Solving Systems of Linear Equations with ConstantCoefficients Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 11.3 Exercises for Independent Work 12 Integrating in Many Dimensions 12.1 Examining the Integrability of Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution 12.2 Calculating Integrals on Given Domains Problem 1 Solution Problem 2 Solution Problem 3 Solution 12.3 Changing the Order of Integration Problem 1 Solution Problem 2 Solution Problem 3 Solution 12.4 Exercises for Independent Work 13 Applying Multidimensional Integrals to Geometry and Physics 13.1 Finding Areas of Flat Surfaces Problem 1 Solution Problem 2 Solution Problem 3 Solution 13.2 Calculating Volumes of Solids Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 13.3 Finding Center-of-Mass Locations Problem 1 Solution Problem 2 Solution Problem 3 Solution 13.4 Calculating Moments of Inertia Problem 1 Solution Problem 2 Solution Problem 3 Solution 13.5 Finding Various Physical Quantities Problem 1 Solution Problem 2 Solution Problem 3 Solution 13.6 Exercises for Independent Work Index "Based on the author's years of teaching experience, this work stands out by providing detailed solutions (often several pages long) to the problems. The basic premise of the book is that no topic should be left unexplained, and no question that could realistically arise while studying the solutions should remain unanswered. The style and format are straightforward and accessible. In addition, each chapter includes exercises for students to work on independently. Answers are provided to all problems, allowing students to check their work"--Provided by the publisher