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.
Line Integrals Part
I Here we will start looking at line
integrals. In particular we will look at
line integrals with respect to arc length.
Line Integrals Part
II We will continue looking at line integrals in
this section. Here we will be looking at
line integrals with respect to x, y, and/or z.
Line Integrals of Vector Fields Here we will look at a third type of line
integrals, line integrals of vector fields.
Fundamental Theorem for Line Integrals In this section we will look at a version of
the fundamental theorem of calculus for line integrals of vector fields.
Conservative Vector Fields Here we will take a somewhat detailed look at
conservative vector fields and how to find potential functions.
Green’s Theorem We will give Green’s Theorem in this section
as well as an interesting application of Green’s Theorem.
Curl and Divergence In this section we will introduce the concepts
of the curl and the divergence of a vector field. We will also give two vector forms of Green’s
Theorem.