In this chapter we will start taking a more detailed look at
three dimensional space (3-D space or ).
This is a very important topic for Calculus III since a good portion of
Calculus III is done in three (or higher) dimensional space.
We will be looking at the equations of graphs in 3-D space
as well as vector valued functions and how we do calculus with them. We will also be taking a look at a couple of
new coordinate systems for 3-D space.
This is the only chapter that exists in two places in my
notes. When I originally wrote these
notes all of these topics were covered in Calculus II however, we have since
moved several of them into Calculus III.
So, rather than split the chapter up I have kept it in the Calculus II
notes and also put a copy in the Calculus III notes. Many of the sections not covered in Calculus
III will be used on occasion there anyway and so they serve as a quick
reference for when we need them.
Here is a list of topics in this chapter.
The
3-D Coordinate System We will introduce the concepts and notation
for the three dimensional coordinate system in this section.
Equations of Lines In this section we will develop the various forms
for the equation of lines in three dimensional space.
Equations of Planes Here we will develop the equation of a plane.
Quadric Surfaces In this section we will be looking at some
examples of quadric surfaces.
Functions of Several Variables A quick review of some important topics about
functions of several variables.
Vector Functions We introduce the concept of vector functions
in this section. We concentrate
primarily on curves in three dimensional space.
We will however, touch briefly on surfaces as well.
Calculus with Vector Functions Here we will take a quick look at limits,
derivatives, and integrals with vector functions.
Tangent, Normal and Binormal Vectors We will define the tangent, normal and
binormal vectors in this section.
Arc Length with Vector Functions In this section we will find the arc length of
a vector function.
Curvature
We will determine the curvature of a function
in this section.
Velocity and Acceleration In this section we will revisit a standard
application of derivatives. We will look
at the velocity and acceleration of an object whose position function is given
by a vector function.
Cylindrical Coordinates We will define the cylindrical coordinate
system in this section. The cylindrical
coordinate system is an alternate coordinate system for the three dimensional
coordinate system.
Spherical Coordinates In this section we will define the spherical
coordinate system. The spherical
coordinate system is yet another alternate coordinate system for the three
dimensional coordinate system.