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2016 | Buch

Results from One-Variable Calculus Kapitel lesen Erstes Kapitel lesen

Calculus and Analysis in Euclidean Space

verfasst von: Jerry Shurman

Verlag: Springer International Publishing

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum. This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis. The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus. More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject.

The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skills of

geometric intuition (the visual cortex being quickly instinctive) algebraic manipulation (symbol-patterns being precise and robust) incisive use of natural language (slogans that encapsulate central ideas enabling a large-scale grasp of the subject).

Thinking in these ways renders mathematics coherent, inevitable, and fluid.

The prerequisite is single-variable calculus, including familiarity with the foundational theorems and some experience with proofs.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Results from One-Variable Calculus
Abstract
This chapter gives a quick review of the real number system and of the foundational and basic theorems of one-variable calculus, and then it proceeds to a more extensive discussion of Taylor’s theorem.
Jerry Shurman
Correction to: Calculus and Analysis in Euclidean Space
Jerry Shurman

Multivariable Differential Calculus

Frontmatter
Chapter 2. Euclidean Space
Abstract
This chapter introduces Euclidean space, discussing its algebra, its geometry, its analysis, and its topology. The main result of the chapter is that the continuous image of a compact set is compact.
Jerry Shurman
Chapter 3. Linear Mappings and Their Matrices
Abstract
This chapter introduces linear mappings and their matrices, working both intrinsically and in coordinates. The determinant is introduced as a function characterized by three properties that have immediate, significant consequences. The magnitude and the sign of the determinant are shown to describe how a linear mapping magnifies volume, and whether it preserves or reverses orientation.
Jerry Shurman
Chapter 4. The Derivative
Abstract
This chapter introduces the multivariable derivative, defined as a linear mapping and characterized intrinsically by the property of closely approximating the original mapping near the point of approximation. The derivative properties are established using the Bachmann–Landau notat–on scheme, allowing a range of ideas from calculus to be handled uniformly. The chapter then computes the derivative in coordinates, and it studies the multivariable counterparts of higher-order derivatives from one-variable calculus. Finally it discusses optimization and directional derivatives.
Jerry Shurman
Chapter 5. Inverse and Implicit Functions
Abstract
This chapter uses the results of the previous three chapters to prove the inverse function theorem, that an invertible derivative connotes a locally invertible mapping. Equivalently, the implicit function theorem states that under some conditions, a set of constraints on a set of variables locally specifies some of the variables as functions of the others. The Lagrange multiplier condition follows, giving a method to solve optimization problems with constraints, i.e., to begin doing calculus in curved spaces.
Jerry Shurman

Multivariable Integral Calculus

Frontmatter
Chapter 6. Integration
Abstract
This chapter introduces the integral of a scalar-valued function of many variables, initially taken over boxes and then over more generally shaped regions. The main tools for evaluating integrals are Fubini’s theorem, which reduces a multivariable integral to a nesting of one-variable integrals, and the change of variable theorem, which replaces one multivariable integral by another that may be easier to evaluate.
Jerry Shurman
Chapter 7. Approximation by Smooth Functions
Abstract
This chapter explains that many functions are closely approximated by smooth functions, meaning functions whose derivatives of all orders exist. The approximation technology is the convolution, essentially integration against a tall, narrow pulse. Having discussed approximation by convolution, we may freely assume from now on that our functions are smooth.
Jerry Shurman
Chapter 8. Parametrized Curves
Abstract
This chapter introduces parametrized curves and discusses their local theory.
Jerry Shurman
Chapter 9. Integration of Differential Forms
Abstract
This chapter covers the integration of differential forms over surfaces, culminating in the general fundamental theorem of integral calculus. The general fundamental theorem is often called Stokes’s theorem, but along with generalizing the classical Stokes’s theorem, it also subsumes the divergence theorem (or Gauss’s theorem), Green’s theorem, and the one-variable fundamental theorem. Much of the chapter’s work is algebraic, in particular the result that differential forms innately pass through changes of variable.
Jerry Shurman
Backmatter
Metadaten
Titel
Calculus and Analysis in Euclidean Space
verfasst von
Jerry Shurman
Copyright-Jahr
2016
Electronic ISBN
978-3-319-49314-5
Print ISBN
978-3-319-49312-1
DOI
https://doi.org/10.1007/978-3-319-49314-5

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