Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Analysis of Functions of a Single Variable » Integration Over Smooth Curves in the Plane C=2π r

Navigation

Table of Contents

Recently Viewed

This feature requires Javascript to be enabled.
 

Integration Over Smooth Curves in the Plane C=2π r

Module by: Lawrence Baggett. E-mail the author

Summary: A brief introduction of the upcoming chapter by Lawrence Baggett. In this chapter we will define what we mean by a smooth curve in the plane and what is meant by its arc length. These definitions are a good bit more tricky than one might imagine. Indeed, it is the subtlety of the definition of arc length that prevented us from defining the trigonometric functions in terms of wrapping the real line around the circle, a definition frequently used in high school trigonometry courses. Having made a proper definition of arc length, we will then be able to establish the formula C=2πrC=2πr for the circumference of a circle of radius r.r.

In this chapter we will define what we mean by a smooth curve in the plane and what is meant by its arc length. These definitions are a good bit more tricky than one might imagine. Indeed, it is the subtlety of the definition of arc length that prevented us from defining the trigonometric functions in terms of wrapping the real line around the circle, a definition frequently used in high school trigonometry courses. Having made a proper definition of arc length, we will then be able to establish the formula C=2πrC=2πr for the circumference of a circle of radius r.r.

By the “plane,” we will mean R2≡C,R2≡C, and we will on occasion want to carefully distinguish between these two notions of the plane, i.e., two real variables xx and yy as opposed to one complex variable z=x+iy.z=x+iy. In various instances, for clarity, we will use notations like x+iyx+iy and (x,y),(x,y), remembering that both of these represent the same point in the plane. As x+iy,x+iy, it is a single complex number, while as (x,y)(x,y) we may think of it as a vector in R2R2 having a magnitude and, if nonzero, a direction.

We also will define in this chapter three different kinds of integrals over such curves. The first kind, called “integration with respect to arc length,” will be completely analogous to the integral defined in (Reference) for functions on a closed and bounded interval, and it will only deal with functions whose domain is the set consisting of the points on the curve. The second kind of integral, called a “contour integral,” is similar to the first one, but it emphasizes in a critical way that we are integrating a complex-valued function over a curve in the complex plane CC and not simply over a subset of R2.R2. The applications of contour integrals is usually to functions whose domains are open subsets of the plane that contain the curve as a proper subset, i.e., whose domains are larger than just the curve. The third kind of integral over a curve, called a “line integral,” is conceptually very different from the first two. In fact, we won't be integrating functions at all but rather a new notion that we call “differential forms.” This is actually the beginnings of the subject called differential geometry, whose intricacies and power are much more evident in higher dimensions than 2.

The main points of this chapter include:

  1. The definition of a smooth curve, and the definition of its arc length,
  2. the derivation of the formula C=2πrC=2πr for the circumference of a circle of radius rr ((Reference)),
  3. the definition of the integral with respect to arc length,
  4. the definition of a contour integral,
  5. the definition of a line integral, and
  6. Green's Theorem ((Reference)).

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks