Calculus III (Math 3435)
Here are my online notes for my Calculus III course that I
teach here at Lamar University. Despite
the fact that these are my “class notes”, they should be accessible to anyone
wanting to learn Calculus III or needing a refresher in some of the topics from
the class.
These notes do assume that the reader has a good working
knowledge of Calculus I topics including limits, derivatives and
integration. It also assumes that the
reader has a good knowledge of several Calculus II topics including some
integration techniques, parametric equations, vectors, and knowledge of three
dimensional space.
Here are a couple of warnings to my students who may be here
to get a copy of what happened on a day that you missed.
- Because
I wanted to make this a fairly complete set of notes for anyone wanting to
learn calculus I have included some material that I do not usually have
time to cover in class and because this changes from semester to semester
it is not noted here. You will need
to find one of your fellow class mates to see if there is something in
these notes that wasn’t covered in class.
- In
general I try to work problems in class that are different from my
notes. However, with Calculus III
many of the problems are difficult to make up on the spur of the moment
and so in this class my class work will follow these notes fairly close as
far as worked problems go. With
that being said I will, on occasion, work problems off the top of my head
when I can to provide more examples than just those in my notes. Also, I often don’t have time in class
to work all of the problems in the notes and so you will find that some
sections contain problems that weren’t worked in class due to time
restrictions.
- Sometimes
questions in class will lead down paths that are not covered here. I try to anticipate as many of the
questions as possible in writing these up, but the reality is that I can’t
anticipate all the questions.
Sometimes a very good question gets asked in class that leads to
insights that I’ve not included here.
You should always talk to someone who was in class on the day you
missed and compare these notes to their notes and see what the differences
are.
- This
is somewhat related to the previous three items, but is important enough
to merit its own item. THESE NOTES
ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!
Using these notes as a substitute for class is liable to get you in
trouble. As already noted not everything in these notes is covered in
class and often material or insights not in these notes is covered in class.
Here is a listing and brief description of the material in
this set of notes.
Three Dimensional
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.
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.
Partial
Derivatives
Limits Taking limits of functions of several
variables.
Partial Derivatives In this section we will introduce the idea of
partial derivatives as well as the standard notations and how to compute them.
Interpretations of Partial Derivatives
Here we will take a look at a couple of
important interpretations of partial derivatives.
Higher Order Partial Derivatives We will take a look at higher order partial
derivatives in this section.
Differentials In this section we extend the idea of
differentials to functions of several variables.
Chain Rule Here we will look at the chain rule for
functions of several variables.
Directional Derivatives We will introduce the concept of directional
derivatives in this section. We will
also see how to compute them and see a couple of nice facts pertaining to
directional derivatives.
Applications
of Partial Derivatives
Tangent Planes and Linear Approximations
We’ll take a look at tangent planes to
surfaces in this section as well as an application of tangent planes.
Gradient Vector, Tangent Planes and Normal
Lines In this section we’ll see how the gradient
vector can be used to find tangent planes and normal lines to a surface.
Relative Minimums and Maximums Here we will see how to identify relative
minimums and maximums.
Absolute Minimums and Maximums We will find absolute minimums and maximums of
a function over a given region.
Lagrange Multipliers In this section we’ll see how to use Lagrange
Multipliers to find the absolute extrema for a function subject to a given constraint.
Multiple
Integrals
Double Integrals We will define the double integral in this
section.
Iterated Integrals In this section we will start looking at how
we actually compute double integrals.
Double Integrals over General Regions
Here we will look at some general double
integrals.
Double Integrals in Polar Coordinates
In this section we will take a look at
evaluating double integrals using polar coordinates.
Triple Integrals Here we will define the triple integral as
well as how we evaluate them.
Triple Integrals in Cylindrical Coordinates
We will evaluate triple integrals using
cylindrical coordinates in this section.
Triple Integrals in Spherical Coordinates
In this section we will evaluate triple
integrals using spherical coordinates.
Change of Variables In this section we will look at change of
variables for double and triple integrals.
Surface Area Here we look at the one real application of
double integrals that we’re going to look at in this material.
Area and Volume Revisited We summarize the area and volume formulas from
this chapter.
Line
Integrals
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.
Surface
Integrals
Parametric Surfaces In this section we will take a look at the
basics of representing a surface with parametric equations. We will also take a look at a couple of
applications.
Surface Integrals Here we will introduce the topic of surface
integrals. We will be working with surface
integrals of functions in this section.
Surface Integrals of Vector Fields We will look at surface integrals of vector
fields in this section.
Stokes’ Theorem We will look at Stokes’ Theorem in this
section.
Divergence Theorem Here we will take a look at the Divergence
Theorem.