# Chapter 16 – Vector Calculus - PowerPoint PPT Presentation

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Chapter 16 – Vector Calculus. 16.9 The Divergence Theorem. Objectives: Understand The Divergence Theorem for simple solid regions. Use Stokes’ Theorem to evaluate integrals. Introduction.

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Chapter 16 – Vector Calculus

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## Chapter 16 – Vector Calculus

16.9 The Divergence Theorem

• Objectives:

• Understand The Divergence Theorem for simple solid regions.

• Use Stokes’ Theorem to evaluate integrals

16.9 The Divergence Theorem

### Introduction

• In Section 16.5, we rewrote Green’s Theorem in a vector version as: where C is the positively oriented boundary curve of the plane region D.

16.9 The Divergence Theorem

### Equation 1

• If we were seeking to extend this theorem to vector fields on 3, we might make the guess that

where S is the boundary surface of the solid region E.

16.9 The Divergence Theorem

### Introduction

• It turns out that Equation 1 is true, under appropriate hypotheses, and is called the Divergence Theorem.

• Notice its similarity to Green’s Theorem and Stokes’ Theorem in that:

• It relates the integral of a derivative of a function (div Fin this case) over a region to the integral of the original function F over the boundary of the region.

16.9 The Divergence Theorem

### Divergence Theorem

• Let:

• E be a simple solid region and let Sbe the boundary surface of E, given with positive (outward) orientation.

• F be a vector field whose component functions have continuous partial derivatives on an open region that contains E.

• Then,

16.9 The Divergence Theorem

### Divergence Theorem

• Thus, the Divergence Theorem states that:

• Under the given conditions, the flux of Facross the boundary surface of E is equal to the triple integral of the divergence ofFoverE.

16.9 The Divergence Theorem

### History

• The Divergence Theorem is sometimes called Gauss’s Theorem after the great German mathematician Karl Friedrich Gauss (1777–1855).

• He discovered this theorem during his investigation of electrostatics.

16.9 The Divergence Theorem

### History

• In Eastern Europe, it is known as Ostrogradsky’s Theorem after the Russian mathematician Mikhail Ostrogradsky

(1801–1862).

• He published this result in 1826.

16.9 The Divergence Theorem

### Example 1

• Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S.

16.9 The Divergence Theorem

### Example 2

• Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S.

16.9 The Divergence Theorem

### Example 3

• Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S.

16.9 The Divergence Theorem

### Example 4

• Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S.

16.9 The Divergence Theorem

### Example 5 – pg. 1157 #11

• Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S.

16.9 The Divergence Theorem

### More Examples

The video examples below are from section 16.9 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.

• Example 1

• Example 2

16.9 The Divergence Theorem

### Demonstrations

• Feel free to explore these demonstrations below.

• The Divergence Theorem

• Vector Field with Sources and Sinks

16.9 The Divergence Theorem

### Review of Chapter

• The main results of this chapter are all higher-dimensional versions of the Fundamental Theorem of Calculus (FTC).

• To help you remember them, we collect them here (without hypotheses) so that you can see more easily their essential similarity.

16.9 The Divergence Theorem

### Review of Chapter

• In each case, notice that:

• On the left side, we have an integral of a “derivative” over a region.

• The right side involves the values of the original function only on the boundaryof the region.

16.9 The Divergence Theorem

### Fundamental Theorem of Calculus

16.9 The Divergence Theorem

### Fundamental Theorem for Line Integrals

16.9 The Divergence Theorem

### Green’s Theorem

16.9 The Divergence Theorem

### Stokes’ Theorem

16.9 The Divergence Theorem

### Divergence Theorem

16.9 The Divergence Theorem