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 Three Dimensional Space

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.

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