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Chapter 5 : Line Integrals

In this section we are going to start looking at Calculus with vector fields (which we’ll define in the first section). In particular we will be looking at a new type of integral, the line integral and some of the interpretations of the line integral. We will also take a look at one of the more important theorems involving line integrals, Green’s Theorem.

Here is a listing of the topics covered in this chapter.

Vector Fields – In this section we introduce the concept of a vector field and give several examples of graphing them. We also revisit the gradient that we first saw a few chapters ago.

Line Integrals – Part I – In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood. We will then formally define the first kind of line integral we will be looking at : line integrals with respect to arc length..

Line Integrals – Part II – In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at : line integrals with respect to \(x\), \(y\), and/or \(z\). We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion.

Line Integrals of Vector Fields – In this section we will define the third type of line integrals we’ll be looking at : line integrals of vector fields. We will also see that this particular kind of line integral is related to special cases of the line integrals with respect to x, y and z.

Fundamental Theorem for Line Integrals – In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed. We will also give quite a few definitions and facts that will be useful.

Conservative Vector Fields – In this section we will take a more detailed look at conservative vector fields than we’ve done in previous sections. We will also discuss how to find potential functions for conservative vector fields.

Green’s Theorem – In this section we will discuss Green’s Theorem as well as an interesting application of Green’s Theorem that we can use to find the area of a two dimensional region.