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This book is devoted to a detailed development of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil variety are involved.

In Part I the divergence theorem is established by a combinatorial argument involving dyadic cubes. Only elementary properties of the Lebesgue integral and Hausdorff measures are used. The resulting integration by parts is sufficiently general for many applications. As an example, it is applied to removable singularities of Cauchy–Riemann, Laplace, and minimal surface equations.

The sets of finite perimeter are introduced in Part II. Both the geometric and analytic points of view are presented. The equivalence of these viewpoints is obtained via the functions of bounded variation. These functions are studied in a self-contained manner with no references to Sobolev’s spaces. The coarea theorem provides a link between the sets of finite perimeter and functions of bounded variation.

The general divergence theorem for bounded vector fields is proved in Part III. The proof consists of adapting the combinatorial argument of Part I to sets of finite perimeter. The unbounded vector fields and mean divergence are also discussed. The final chapter contains a characterization of the distributions that are equal to the flux of a continuous vector field.

**DYADIC FIGURES**

**Preliminaries **

The setting

Topology

Measures

Hausdorff measures

Differentiable and Lipschitz maps

**Divergence Theorem for Dyadic Figures **

Differentiable vector fields

Dyadic partitions

Admissible maps

Convergence of dyadic figures

**Removable Singularities **

Distributions

Differential equations

Holomorphic functions

Harmonic functions

The minimal surface equation

Injective limits

**SETS OF FINITE PERIMETER**

**Perimeter **

Measure-theoretic concepts

Essential boundary

Vitali’s covering theorem

Density

Definition of perimeter

Line sections

**BV Functions **

Variation

Mollification

Vector valued measures

Weak convergence

Properties of BV functions

Approximation theorem

Coarea theorem

Bounded convex domains

Inequalities

**Locally BV Sets **

Dimension one

Besicovitch’s covering theorem

The reduced boundary

Blow-up

Perimeter and variation

Properties of BV sets

Approximating by figures

**THE DIVERGENCE THEOREM **

**Bounded Vector Fields**

Approximating from inside

Relative derivatives

The critical interior

The divergence theorem

Lipschitz domains

**Unbounded Vector Fields**

Minkowski contents

Controlled vector fields

Integration by parts

**Mean Divergence **

The derivative

The critical variation

**Charges **

Continuous vector fields

Localized topology

Locally convex spaces

Duality

The space *BV _{c}*(Ω)

Streams

**The Divergence Equation **

Background

Solutions in *L ^{p}*(Ω; R

^{n})

Continuous solutions

**Bibliography **

**List of Symbols **

**Index**

This book is devoted to a detailed development of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil variety are involved.

In Part I the divergence theorem is established by a combinatorial argument involving dyadic cubes. Only elementary properties of the Lebesgue integral and Hausdorff measures are used. The resulting integration by parts is sufficiently general for many applications. As an example, it is applied to removable singularities of Cauchy–Riemann, Laplace, and minimal surface equations.

The sets of finite perimeter are introduced in Part II. Both the geometric and analytic points of view are presented. The equivalence of these viewpoints is obtained via the functions of bounded variation. These functions are studied in a self-contained manner with no references to Sobolev’s spaces. The coarea theorem provides a link between the sets of finite perimeter and functions of bounded variation.

The general divergence theorem for bounded vector fields is proved in Part III. The proof consists of adapting the combinatorial argument of Part I to sets of finite perimeter. The unbounded vector fields and mean divergence are also discussed. The final chapter contains a characterization of the distributions that are equal to the flux of a continuous vector field.

**DYADIC FIGURES**

**Preliminaries **

The setting

Topology

Measures

Hausdorff measures

Differentiable and Lipschitz maps

**Divergence Theorem for Dyadic Figures **

Differentiable vector fields

Dyadic partitions

Admissible maps

Convergence of dyadic figures

**Removable Singularities **

Distributions

Differential equations

Holomorphic functions

Harmonic functions

The minimal surface equation

Injective limits

**SETS OF FINITE PERIMETER**

**Perimeter **

Measure-theoretic concepts

Essential boundary

Vitali’s covering theorem

Density

Definition of perimeter

Line sections

**BV Functions **

Variation

Mollification

Vector valued measures

Weak convergence

Properties of BV functions

Approximation theorem

Coarea theorem

Bounded convex domains

Inequalities

**Locally BV Sets **

Dimension one

Besicovitch’s covering theorem

The reduced boundary

Blow-up

Perimeter and variation

Properties of BV sets

Approximating by figures

**THE DIVERGENCE THEOREM **

**Bounded Vector Fields**

Approximating from inside

Relative derivatives

The critical interior

The divergence theorem

Lipschitz domains

**Unbounded Vector Fields**

Minkowski contents

Controlled vector fields

Integration by parts

**Mean Divergence **

The derivative

The critical variation

**Charges **

Continuous vector fields

Localized topology

Locally convex spaces

Duality

The space *BV _{c}*(Ω)

Streams

**The Divergence Equation **

Background

Solutions in *L ^{p}*(Ω; R

^{n})

Continuous solutions

**Bibliography **

**List of Symbols **

**Index**

This book is devoted to a detailed development of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil variety are involved.

In Part I the divergence theorem is established by a combinatorial argument involving dyadic cubes. Only elementary properties of the Lebesgue integral and Hausdorff measures are used. The resulting integration by parts is sufficiently general for many applications. As an example, it is applied to removable singularities of Cauchy–Riemann, Laplace, and minimal surface equations.

The sets of finite perimeter are introduced in Part II. Both the geometric and analytic points of view are presented. The equivalence of these viewpoints is obtained via the functions of bounded variation. These functions are studied in a self-contained manner with no references to Sobolev’s spaces. The coarea theorem provides a link between the sets of finite perimeter and functions of bounded variation.

The general divergence theorem for bounded vector fields is proved in Part III. The proof consists of adapting the combinatorial argument of Part I to sets of finite perimeter. The unbounded vector fields and mean divergence are also discussed. The final chapter contains a characterization of the distributions that are equal to the flux of a continuous vector field.

**DYADIC FIGURES**

**Preliminaries **

The setting

Topology

Measures

Hausdorff measures

Differentiable and Lipschitz maps

**Divergence Theorem for Dyadic Figures **

Differentiable vector fields

Dyadic partitions

Admissible maps

Convergence of dyadic figures

**Removable Singularities **

Distributions

Differential equations

Holomorphic functions

Harmonic functions

The minimal surface equation

Injective limits

**SETS OF FINITE PERIMETER**

**Perimeter **

Measure-theoretic concepts

Essential boundary

Vitali’s covering theorem

Density

Definition of perimeter

Line sections

**BV Functions **

Variation

Mollification

Vector valued measures

Weak convergence

Properties of BV functions

Approximation theorem

Coarea theorem

Bounded convex domains

Inequalities

**Locally BV Sets **

Dimension one

Besicovitch’s covering theorem

The reduced boundary

Blow-up

Perimeter and variation

Properties of BV sets

Approximating by figures

**THE DIVERGENCE THEOREM **

**Bounded Vector Fields**

Approximating from inside

Relative derivatives

The critical interior

The divergence theorem

Lipschitz domains

**Unbounded Vector Fields**

Minkowski contents

Controlled vector fields

Integration by parts

**Mean Divergence **

The derivative

The critical variation

**Charges **

Continuous vector fields

Localized topology

Locally convex spaces

Duality

The space *BV _{c}*(Ω)

Streams

**The Divergence Equation **

Background

Solutions in *L ^{p}*(Ω; R

^{n})

Continuous solutions

**Bibliography **

**List of Symbols **

**Index**

**DYADIC FIGURES**

**Preliminaries **

The setting

Topology

Measures

Hausdorff measures

Differentiable and Lipschitz maps

**Divergence Theorem for Dyadic Figures **

Differentiable vector fields

Dyadic partitions

Admissible maps

Convergence of dyadic figures

**Removable Singularities **

Distributions

Differential equations

Holomorphic functions

Harmonic functions

The minimal surface equation

Injective limits

**SETS OF FINITE PERIMETER**

**Perimeter **

Measure-theoretic concepts

Essential boundary

Vitali’s covering theorem

Density

Definition of perimeter

Line sections

**BV Functions **

Variation

Mollification

Vector valued measures

Weak convergence

Properties of BV functions

Approximation theorem

Coarea theorem

Bounded convex domains

Inequalities

**Locally BV Sets **

Dimension one

Besicovitch’s covering theorem

The reduced boundary

Blow-up

Perimeter and variation

Properties of BV sets

Approximating by figures

**THE DIVERGENCE THEOREM **

**Bounded Vector Fields**

Approximating from inside

Relative derivatives

The critical interior

The divergence theorem

Lipschitz domains

**Unbounded Vector Fields**

Minkowski contents

Controlled vector fields

Integration by parts

**Mean Divergence **

The derivative

The critical variation

**Charges **

Continuous vector fields

Localized topology

Locally convex spaces

Duality

The space *BV _{c}*(Ω)

Streams

**The Divergence Equation **

Background

Solutions in *L ^{p}*(Ω; R

^{n})

Continuous solutions

**Bibliography **

**List of Symbols **

**Index**

**DYADIC FIGURES**

**Preliminaries **

The setting

Topology

Measures

Hausdorff measures

Differentiable and Lipschitz maps

**Divergence Theorem for Dyadic Figures **

Differentiable vector fields

Dyadic partitions

Admissible maps

Convergence of dyadic figures

**Removable Singularities **

Distributions

Differential equations

Holomorphic functions

Harmonic functions

The minimal surface equation

Injective limits

**SETS OF FINITE PERIMETER**

**Perimeter **

Measure-theoretic concepts

Essential boundary

Vitali’s covering theorem

Density

Definition of perimeter

Line sections

**BV Functions **

Variation

Mollification

Vector valued measures

Weak convergence

Properties of BV functions

Approximation theorem

Coarea theorem

Bounded convex domains

Inequalities

**Locally BV Sets **

Dimension one

Besicovitch’s covering theorem

The reduced boundary

Blow-up

Perimeter and variation

Properties of BV sets

Approximating by figures

**THE DIVERGENCE THEOREM **

**Bounded Vector Fields**

Approximating from inside

Relative derivatives

The critical interior

The divergence theorem

Lipschitz domains

**Unbounded Vector Fields**

Minkowski contents

Controlled vector fields

Integration by parts

**Mean Divergence **

The derivative

The critical variation

**Charges **

Continuous vector fields

Localized topology

Locally convex spaces

Duality

The space *BV _{c}*(Ω)

Streams

**The Divergence Equation **

Background

Solutions in *L ^{p}*(Ω; R

^{n})

Continuous solutions

**Bibliography **

**List of Symbols **

**Index**

**DYADIC FIGURES**

**Preliminaries **

The setting

Topology

Measures

Hausdorff measures

Differentiable and Lipschitz maps

**Divergence Theorem for Dyadic Figures **

Differentiable vector fields

Dyadic partitions

Admissible maps

Convergence of dyadic figures

**Removable Singularities **

Distributions

Differential equations

Holomorphic functions

Harmonic functions

The minimal surface equation

Injective limits

**SETS OF FINITE PERIMETER**

**Perimeter **

Measure-theoretic concepts

Essential boundary

Vitali’s covering theorem

Density

Definition of perimeter

Line sections

**BV Functions **

Variation

Mollification

Vector valued measures

Weak convergence

Properties of BV functions

Approximation theorem

Coarea theorem

Bounded convex domains

Inequalities

**Locally BV Sets **

Dimension one

Besicovitch’s covering theorem

The reduced boundary

Blow-up

Perimeter and variation

Properties of BV sets

Approximating by figures

**THE DIVERGENCE THEOREM **

**Bounded Vector Fields**

Approximating from inside

Relative derivatives

The critical interior

The divergence theorem

Lipschitz domains

**Unbounded Vector Fields**

Minkowski contents

Controlled vector fields

Integration by parts

**Mean Divergence **

The derivative

The critical variation

**Charges **

Continuous vector fields

Localized topology

Locally convex spaces

Duality

The space *BV _{c}*(Ω)

Streams

**The Divergence Equation **

Background

Solutions in *L ^{p}*(Ω; R

^{n})

Continuous solutions

**Bibliography **

**List of Symbols **

**Index**