Extra Credit Offer: If you find an error in any handout, let me know. If you are the first to report it, you will get some extra credit on quizzes. This includes errors in spelling and grammar, but mathematical errors will be worth more.
Important:
Expectations
for
group work and writing.
Exams from previous semesters.
How to Think about Points and Vectors Without Coordinates
Cylindrical and Spherical
Coordinates for Mathematicians (as Opposed to Physicists)
Two Useful Things from Linear Algebra
* Finding Coefficients Relative to an Orthogonal Basis
* Finding the matrix of a linear
transformation
Informal Derivation of Arc Length Formulas
Geometry of Curves in E2
For the geometry of curves topic, this handout will be our main
text.
You should still read Section 3.2 of Colley for additional
insight, but
the computations I expect you to do will be done as they are in
the
handout. Be sure to read about this section in the "Comments on
our
Text" handout.
Curves and Surfaces
This is a pdf made from a Mathematica notebook. The
notebook is
what you should really go through, as it has several graphics you
can
manipulate. The notebook is in the Courses on Caleb folder
N:\Math\Math225.
Derivatives Along Vectors and Directional
Derivatives
Read this in three pieces.
1) The first two sections, "Derivatives Along
Vectors" and "Directional Derivatives" through the bottom of page
4.
2) "Derivatives Along Vectors for Mappings"
pages 5
and 6.
3) "The Chain Rule and Linear Algebra to the
Rescue!" pages 7 and 8.
The Idea of Differentiability for
Functions of Several Variables
This goes with Section 2.3 of Colley.
When is a Vector Field the
Gradient of a Function?
Insert for Colley's text, Sections 3.3 and 3.4.
The Curl Operator in Two Dimensions
Section 3.4 of Colley's text.
Gradient, Divergence, Laplacian, and
Curl
in Other Coordinate Systems
This goes with Section
3.4 of Colley.
Steepest Descent
This is an important numerical technique for approximating the
minimum
of a function of several variables.
Path Integrals With Respect to Arc
Length and Problems
This goes with Section 6.1 in Colley.
Summary of Path Integrals
Chapter 6 of Colley introduces path integrals and explores their
consequences. This pulls together a lot of things we have done so
far,
and introduces several new ideas as well. It can be a bit
overwhelming.
This is a summary.
A Curl-Free Vector Field
that
is not a Gradient
It is easy to think that if curl F = 0, then F
must be a gradient (or be conservative), but it's not always the
case.
It's almost true, and this shows what the possibilities are in R2.
Two papers of mine that you can
read based on things you learned in this class.
Regularity of the
Distance
Function
Suppose that M is a Ck surface in R3 (this
means
that M is locally the graph of a Ck function of 2
variables). Then the unit normal vector field N along the
surface
M is Ck-1 because you have to take first partial
derivatives
to compute N. Let d be the distance function for M, that is,
if X
is a point in R3, then d(X) is the distance from X to
M. You can write a formula near M for d that involves N, and
so
one would expect that d is Ck-1 just as N is. In
fact,
d is Ck. This is not a new result, but it is a
particularly clean, coordinate-free proof. It uses the
Inverse
Function Theorem and directional derivatives.
The Volume Swept Out by a
Moving Planar Region
A surface moving through space sweeps out a volume. This
paper
gives an integral formula for the volume. This is not a new
result, but the only place I have seen it is in a calculus book
from
1934, and it is stated there without proof! This requires
the
change of variables formula for multiple integrals.