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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.

Inhaltsverzeichnis

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

Abstract
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

Abstract
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

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