3. Double Integrals and Line Integrals in the Plane

Unit 3 Introduction

This unit starts our study of integration of functions of several variables. To keep the visualization difficulties to a minimum we will only look at functions of two variables. (We will look at functions of three variables in the next unit.)

Our main objects of study will be two types of integrals:

Double integrals, which are integrals over planar regions.
Line or path integrals, which are integrals over curves.

All integrals can be thought of as a sum, technically a limit of Riemann sums, and these will be no exception. If you make sure you master this simple idea then you will find the applications and proofs involving these integrals to be straightforward.

We will conclude the unit by learning Green’s theorem which relates the two types of integrals and is a generalization of the Fundamental Theorem of Calculus. Along the way we will introduce the concepts of work and two dimensional flux and also two types of derivatives of vector valued functions of two variables, the curl and the divergence.

Part A: Double Integrals

» Session 47: Definition of Double Integration
» Session 48: Examples of Double Integration
» Session 49: Exchanging the Order of Integration
» Session 50: Double Integrals in Polar Coordinates
» Session 51: Applications: Mass and Average Value
» Session 52: Applications: Moment of Inertia
» Session 53: Change of Variables
» Session 54: Example: Polar Coordinates
» Session 55: Example
» Problem Set 7

Part B: Vector Fields and Line Integrals

» Session 56: Vector Fields
» Session 57: Work and Line Integrals
» Session 58: Geometric Approach
» Session 59: Example: Line Integrals for Work
» Session 60: Fundamental Theorem for Line Integrals
» Session 61: Conservative Fields, Path Independence, Exact Differentials
» Session 62: Gradient Fields
» Session 63: Potential Functions
» Session 64: Curl
» Problem Set 8

Part C: Green’s Theorem

» Session 65: Green’s Theorem
» Session 66: Curl(F) = 0 Implies Conservative
» Session 67: Proof of Green’s Theorem
» Session 68: Planimeter: Green’s Theorem and Area
» Session 69: Flux in 2D
» Session 70: Normal Form of Green’s Theorem
» Session 71: Extended Green’s Theorem: Boundaries with Multiple Pieces
» Session 72: Simply Connected Regions and Conservative
» Problem Set 9

Exam 3

» Practice Exam
» Session 73: Exam Review
» Exam

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» Practice Exam
» Session 73: Exam Review
» Exam

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In part A, we will learn about double integration over regions in the plane. Conceptually an integral is a sum. We will apply this idea to computing the mass, center of mass and moment of inertia of a two dimensional body and the volume of a region bounded by surfaces.

In order to compute double integrals we will have to describe regions in the plane in terms of the equations describing their boundary curves. After that, the computation just becomes two single variable integrations done iteratively.

» Session 47: Definition of Double Integration
» Session 48: Examples of Double Integration
» Session 49: Exchanging the Order of Integration
» Session 50: Double Integrals in Polar Coordinates
» Session 51: Applications: Mass and Average Value
» Session 52: Applications: Moment of Inertia
» Session 53: Change of Variables
» Session 54: Example: Polar Coordinates
» Session 55: Example
» Problem Set 7

« Previous | Next »

A vector field attaches a vector to each point. For example, the sun has a gravitational field, which gives its gravitational attraction at each point in space. The field does work as it moves a mass along a curve. We will learn to express this work as a line integral and to compute its value.

In physics, some force fields conserve energy. Such conservative fields are determined by their potential energy functions. We will define what a conservative field is mathematically and learn to identify them and find their potential function.

» Session 56: Vector Fields
» Session 57: Work and Line Integrals
» Session 58: Geometric Approach
» Session 59: Example: Line Integrals for Work
» Session 60: Fundamental Theorem for Line Integrals
» Session 61: Conservative Fields, Path Independence, Exact Differentials
» Session 62: Gradient Fields
» Session 63: Potential Functions
» Session 64: Curl
» Problem Set 8

« Previous | Next »

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.

» Session 65: Green’s Theorem
» Session 66: Curl(F) = 0 Implies Conservative
» Session 67: Proof of Green’s Theorem
» Session 68: Planimeter: Green’s Theorem and Area
» Session 69: Flux in 2D
» Session 70: Normal Form of Green’s Theorem
» Session 71: Extended Green’s Theorem: Boundaries with Multiple Pieces
» Session 72: Simply Connected Regions and Conservative
» Problem Set 9

« Previous | Next »

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Multivariable Calculus