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2016 | Book

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Multivariable Dynamic Calculus on Time Scales

Authors: Martin Bohner, Svetlin G. Georgiev

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book offers the reader an overview of recent developments of multivariable dynamic calculus on time scales, taking readers beyond the traditional calculus texts. Covering topics from parameter-dependent integrals to partial differentiation on time scales, the book’s nine pedagogically oriented chapters provide a pathway to this active area of research that will appeal to students and researchers in mathematics and the physical sciences. The authors present a clear and well-organized treatment of the concept behind the mathematics and solution techniques, including many practical examples and exercises.

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Table of Contents

Frontmatter

Chapter 1. Time Scales

Abstract

A time scale is an arbitrary nonempty closed subset of the real numbers.

Martin Bohner, Svetlin G. Georgiev

Chapter 2. Differential Calculus of Functions of One Variable

Abstract

Assume that \(f:{\mathbb {T}}\rightarrow {\mathbb {R}}\) is a function and let \(t\in {\mathbb {T}}^{\kappa }\).

Martin Bohner, Svetlin G. Georgiev

Chapter 3. Integral Calculus of Functions of One Variable

A function \(f:{\mathbb {T}}\rightarrow {\mathbb {R}}\) is called regulated provided its right-sided limits exist (finite) at all right-dense points in \({\mathbb {T}}\) and its left-sided limits exist (finite) at all left-dense points in \({\mathbb {T}}\).

Martin Bohner, Svetlin G. Georgiev

Chapter 4. Sequences and Series of Functions

Abstract

Suppose that \(f_n:{\mathbb {T}}\rightarrow {\mathbb {R}}\), \(n\in {\mathbb {N}}\), \(S\subset {\mathbb {T}}\).

Martin Bohner, Svetlin G. Georgiev

Chapter 5. Parameter-Dependent Integrals

Abstract

Let \({\mathbb {T}}_1\) and \({\mathbb {T}}_2\) be time scales.

Martin Bohner, Svetlin G. Georgiev

Chapter 6. Partial Differentiation on Time Scales

Let \(n\in {\mathbb {N}}\) be fixed. For each \(i\in \{1,2,\ldots ,n\}\), we denote by \({\mathbb {T}}_i\) a time scale.

Martin Bohner, Svetlin G. Georgiev

Chapter 7. Multiple Integration on Time Scales

Abstract

Let \({\mathbb {T}}_i\), \(i\in \{1,2,\ldots ,n\}\), be time scales. For \(i\in \{1,2,\ldots ,n\}\), let \(\sigma _i\), \(\rho _i\), and \(\varDelta _i\) denote the forward jump operator, the backward jump operator, and the delta differentiation, respectively, on \({\mathbb {T}}_i\).

Martin Bohner, Svetlin G. Georgiev

Chapter 8. Line Integrals

Abstract

Let \({\mathbb {T}}\) be a time scale with the forward jump operator \(\sigma \) and the delta operator \(\varDelta \). Let \(a,b\in {\mathbb {T}}\) with \(a<b\). Assume that \(\phi _i:[a,b]\rightarrow {\mathbb {R}}\) is continuous, \(i\in \{1,\ldots ,m\}\).

Martin Bohner, Svetlin G. Georgiev

Chapter 9. Surface Integrals

Abstract

Let \({\mathbb {T}}_i\), \(i\in \{1,\ldots , n\}\), be time scales. Suppose \(\varOmega \subset {\mathbb {T}}_1\times \ldots \times {\mathbb {T}}_n\). Let \(\phi _i:\varOmega \rightarrow {\mathbb {R}}\) be continuous functions on \(\varOmega \).

Martin Bohner, Svetlin G. Georgiev

Backmatter

Title: Multivariable Dynamic Calculus on Time Scales
Authors: Martin Bohner
Svetlin G. Georgiev
Copyright Year: 2016
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-47620-9
Print ISBN: 978-3-319-47619-3
DOI: https://doi.org/10.1007/978-3-319-47620-9

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