The3DCoordinateSystem
Well start the chapter off with a fairly short discussion introducing the 3-D coordinate system
and the conventions that well be using. We will also take a brief look at how the different
coordinate systems can change the graph of an
CalculuswithVectorFunctions
In this section we need to talk briefly about limits, derivatives and integrals of vector functions.
As you will see, these behave in a fairly predictable manner. We will be doing all of the work in
but we can naturally extend
ArcLengthwithVectorFunctions
In this section well recast an old formula into terms of vector functions. We want to determine
the length of a vector function,
on the interval
.
We actually already know how to do this. Recall that we can write the vector fu
Tangent,NormalandBinormalVectors
In this section we want to look at an application of derivatives for vector functions. Actually,
there are a couple of applications, but they all come back to needing the first one.
In the past weve used the fact that the
VectorFunctions
We first saw vector functions back when we were looking at the Equation of Lines. In that
section we talked about them because we wrote down the equation of a line in
in
terms of a vector function (sometimes called a vector-valued function
QuadricSurfaces
In the previous two sections weve looked at lines and planes in three dimensions (or
) and while these are used quite heavily at times in a Calculus class there are many other
surfaces that are also used fairly regularly and so we need to
EquationsofPlanes
In the first section of this chapter we saw a couple of equations of planes. However, none of
those equations had three variables in them and were really extensions of graphs that we could
look at in two dimensions. We would like a more
FunctionsofSeveralVariables
In this section we want to go over some of the basic ideas about functions of more than one
variable.
First, remember that graphs of functions of two variables,
are surfaces in three dimensional space. For example here is the g
EquationsofLines
In this section we need to take a look at the equation of a line in
. As we saw in the
previous section the equation
does not describe a
line in
, instead it describes a plane. This doesnt mean however that we cant write
down an equation
Curvature
In this section we want to briefly discuss the curvature of a smooth curve (recall that for a
smooth curve we require
is continuous and
). The curvature measures how fast a curve is changing direction at a given point.
There are several formulas