18.02SC | Fall 2010 | Undergraduate

# Multivariable Calculus

3. Double Integrals and Line Integrals in the Plane

## Part C: Green's Theorem

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In this part we will learn Green’s theorem, which relates line integrals over a closed path to a double integral over the region enclosed. The line integral involves a vector field and the double integral involves derivatives (either div or curl, we will learn both) of the vector field.

First we will give Green’s theorem in work form. The line integral in question is the work done by the vector field. The double integral uses the curl of the vector field. Then we will study the line integral for flux of a field across a curve. Finally we will give Green’s theorem in flux form. This relates the line integral for flux with the divergence of the vector field.

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### Overview

In this session you will:

• Do practice problems
• Use the solutions to check your work

### Problem Set

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Green’s Theorem

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Green’s Theorem (PDF)

### Examples

Using Green’s Theorem (PDF)

### Problems and Solutions

Problems: Green’s Theorem (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes

### Lecture Video

#### Video Excerpts

Clip: Curl(F)=0 Implies Conservative

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Green’s Theorem and Conservative Fields (PDF)

### Problems and Solutions

Problems: Identifying Gradient Fields and Exact Differentials (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Watch two recitation videos

### Lecture Video

#### Video Excerpts

Clip: Planimeter: Green’s Theorem and Area

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Examples

Finding Area Using Line Integrals (PDF)

### Problems and Solutions

Problems: Green’s Theorem and Area (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch lecture video clips and read board notes
• Read course notes and examples
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Flux Across a Curve

Clip: Calculating Flux

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Reading and Examples

Two-Dimensional Flux (PDF)

### Problems and Solutions

Problems: Flux Across Curves (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Introduction and Definition of Divergence

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Green’s Theorem in Normal Form (PDF)

### Examples

Verify Green’s Theorem in Normal Form (PDF)

### Problems and Solutions

Problems: Normal Form of Green’s Theorem (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Extended Green’s Theorem

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Examples

Extended Green’s Theorem (PDF)

### Problems and Solutions

Problems: Extended Green’s Theorem (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Simply Connected Regions and Conservative Fields

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Reading and Examples

Simply-Connected Regions (PDF)

### Problems and Solutions

Problems: Simply Connected Regions and Conservative Fields (PDF)

Solutions (PDF)

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## Course Info

Fall 2010
##### Learning Resource Types
Simulations
Exams with Solutions
Problem Sets with Solutions
Lecture Videos
Lecture Notes