681 pages · 2024
Vector Calculus
By Steven G. Krantz · Harold Parks
- Published
- May 28, 2024
- Imprint
- Chapman and Hall/CRC
- Edition
- 1st Edition
- ISBN
- 9781003304241
Overview
About this book
Using meaningful examples, credible applications, and incisive technology, Vector Calculus strives to empower students, enhance their critical thinking skills, and equip them with the knowledge and skills to succeed in the major or discipline they ultimately choose to study. This text is intended to be a cornerstone of that process. An engaging style and clear writing make the language of mathematics accessible, understandable, and enjoyable, with a high standard for mathematical rigor.
A calculus book must tell the truth. This book is carefully written in the accepted language of mathematics in a readable exposition. It includes useful and fascinating applications, acquaints students with the history of the subject, and offers a sense of what mathematics is all about.
Technique is presented, yet so are ideas. The authors help students to master basic methods and discover and build their own concepts in a scientific subject. There is an emphasis on using modeling and numerical calculation.
Additional features include:
A Quick Quiz and Problems for Practice, Further Theory and Practice, and Calculator/Computer Exercises appear at the end of each section All exercise sets are step laddered A Look Back and A Look Forward help students put the ideas in context Every chapter ends with a Genesis and Development section, giving history and perspective on key topics in the evolution of calculus Boxed Insights clear up points or answer commonly asked questions The text has an extra-large offering of examples Examples are illustrated with meaningful and useful graphics
The pedagogical features make the subject more interesting and accessible to students than other texts, while maintaining an appropriate rigor. —Daniel Cunningham, CSU-Fresno
This text is truly well written and organized. I do like the fact the book is quite rigorous, yet full of illustrative examples. —Bob Devaney, Boston University
Subjects
Engineering & Technology, Mathematics & Statistics
Inside the book
Table of contents
Select a chapter to read its abstract.
Chapter Chapter 1Vectors
Pages: 104
In this chapter, we lay the geometric foundation for all of our coming work with functions of two or more variables. Those studies will require us to have a mathematical model of three-dimensional space. Just as we use a real number as the mathematical model of a point on a line and an ordered pair of real numbers as the mathematical model of a point in the plane, so we use an ordered triple https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math1_1.tif"/> as the mathematical model of a point in three-dimensional space. Equations among these space variables describe curves and surfaces in space. Our first goal is to study the simplest such objects—lines and planes—in detail. As an aid to our investigations, we begin with a new concept, the vector, which will be a vital tool in every chapter to follow.
Chapter Chapter 2Vector-Valued Functions
Pages: 102
Until now, all of the functions https://www.w3.org/1998/Math/MathML" display="inline"> x ↦ f ( x ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_1.tif"/> that we have studied have been scalar valued: We evaluate f at a real number x, and we obtain a real number https://www.w3.org/1998/Math/MathML" display="inline"> f ( x ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_2.tif"/> as the result. In this chapter, we study the calculus of vector-valued functions https://www.w3.org/1998/Math/MathML" display="inline"> t ↦ r ( t ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_3.tif"/> . Here we evaluate a function https://www.w3.org/1998/Math/MathML" display="inline"> r https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_4.tif"/> at a real number t, and we obtain a vector https://www.w3.org/1998/Math/MathML" display="inline"> r ( t ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_5.tif"/> as the result. By writing out the vector https://www.w3.org/1998/Math/MathML" display="inline"> r ( t ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_6.tif"/> in terms of its components https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ x ( t ) , y ( t ) , z ( t ) ⟩ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_7.tif"/> , we see that we can identify the value of https://www.w3.org/1998/Math/MathML" display="inline"> r ( t ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_8.tif"/> with the point https://www.w3.org/1998/Math/MathML" display="inline"> ( x ( t ) , y ( t ) , z ( t ) ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_9.tif"/> in space. As t varies in the domain of https://www.w3.org/1998/Math/MathML" display="inline"> r https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_10.tif"/> , the values https://www.w3.org/1998/Math/MathML" display="inline"> r ( t ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math2_11.tif"/> trace out a curve in three-dimensional space. By developing the calculus of vector-valued functions, we will be able to analyze such space curves.
Chapter Chapter 3Functions of Several Variables
Pages: 164
Many of the quantities that we study in mathematics and other fields depend on two or more variables. For example, the output of a factory depends on the amount of capital allocated to labor and the amount allocated to equipment. The air pressure at a point in the atmosphere depends on the altitude of the point and on the temperature at the point. The height above sea level of a point on Earth's surface depends on the longitude and latitude of the point. Functions of two or more variables are, indeed, commonplace. To analyze these functions, we need to develop appropriate tools of calculus. That is the purpose of this chapter. We will ask and answer the same types of questions that we posed for functions of one variable.
Chapter Chapter 4Multiple Integrals
Pages: 132
In Chapter 3, we learned how to differentiate functions of two or more variables. Now we will learn how to integrate such functions. To be specific, given a region https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math4_1.tif"/> of the xy-plane and a continuous function f defined on https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math4_2.tif"/> , we will define the integral of f over https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math4_3.tif"/> . This integral is denoted by https://www.w3.org/1998/Math/MathML" display="inline"> ∬ R f ( x , y ) dA https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math4_4.tif"/> and is called a double integral. Analogously, if F is a continuous function of three variables that is defined on a solid region https://www.w3.org/1998/Math/MathML" display="inline"> U https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math4_5.tif"/> in xyz-space, then we will define the triple integral https://www.w3.org/1998/Math/MathML" display="inline"> ∭ U F ( x , y , z ) d V https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304241/38285f7e-d7bf-4349-8a62-5a63d13da3b0/content/math4_6.tif"/> . The main theorem of this chapter tells us that multiple integrals can be evaluated by iterating the familiar one-variable process of integration, one variable at a time.
Chapter Chapter 5Vector Calculus
Pages: 146
Imagine a fluid flowing through a planar region. Since velocity has both magnitude and direction, the velocity of the flow at various points can be indicated by vectors. In this chapter, we will learn to relate the flow through the boundary of the region to the flow in the interior of the region. We will see that these physical considerations are related to a remarkable multidimensional version of the fundamental theorem of calculus.