Index of /~mate/misc
Most of these notes were written as supplementary material for
courses I have taught.
has the title "Axiom of completeness," and it discusses
the completeness of the real line. This is part of a
larger set of notes
on elementary real analysis, discussing convergence theorems,
differentiation, and Riemann integration, entitled
"Supplementary notes on introduction to analysis."
has the title "The natural exponential function," and it
discusses an approach to the function exp(x) that does
not rely on integration.
has the tile "Hyperbolic functions," and it contains an
introduction to hyperbolic functions and their relationship
to trigonometric functions. When doing so, it discusses
exponentiation with imaginary exponents that may have
more intuitive appeal than the standard discussion involving
is a set of notes entitled "The Laplace operator in polar
coordinates in several dimensions." It discusses the analogs
of polar and spherical coordinates in higher dimensions,
and discusses the Laplace operator in these coordinates.
is a set of notes with the title "On the equality of
mixed partial derivatives" that describes Peano's
example where to mixed partial derivatives are not
equal, and then discusses a number of conditions, some
less well known, ensuring the equality of mixed
discusses the limit of sin t/t at t=0.
has the title "The remainder term in Taylor's formula"; it
discusses various forms of the remainder term of the Taylor
This is part of a
larger set of notes
on elementary real analysis just mentioned.
is a set of notes entitled "Totally bounded spaces," and it
shows that a metric space is compact if and only if it is
totally bounded and complete.
is a note with title
"The Cayley--Hamilton Theorem" proves Cayley--Hamilton
Theorem using the adjugate (classic adjoint) of a
matrix. It includes a short discussion of formal
polynomials and the evaluation operator.
is a note with title
"The cyclic decomposition theorem" proves the cyclic
decomposition theorem for a vector space over an
arbitrary field, and discusses some consequences,
including the Jordan canonical form of a matrix,
and the square root of a positive matrix.
is a note entitled
"The uniqueness of the row echelon form"
shows that the row echelon form of a matrix is unique.
is a note with title "The Jordan canonical form" proves
the existence of the Jordan canonical form of a matrix
over an algebraically closed field. It gives an example
finding the minimal polynomial of a matrix, and uses
it to find its Jordan canonical form.
is a note entitled "The rotation of a coordinate system
as a linear transformation" uses the theory of linear
transformations to discuss rotations of the coordinate
system in the plane.
is a note with title "The row space of a matrix" shows that
two matrices with the same rowspace have the same row
echelon form, implying the uniqueness of the row echelon
is a note entitled "Determinants." and it gives an introduction
to the theory of determinants. This is part of a larger set of
with the title
"Introduction to numerical analysis with C programs,"
intended as a complete set of notes for a one-semester
course in numerical analysis.
Partial differential equations
has the title "characteristic manifolds of linear partial
discusses the Cauchy problem for higher order linear
partial differential equations, and describes the
significance of the characteristic manifold (not to be
confused with characteristic curves used in solving
first order partial differential equations).
is a note with title "The first integral of a homogeneous
linear partial differential equation," and it discusses
some simple examples to find the first integral.
has the title "First order partial differential equations,"
and discusses the basic theory of first order partial
differential equations, describes characteristic curves
and the complete integral, with some examples.
entitled "Charpit's method to find the complete integral"
describes a method to find the complete integral of a
first order partial differential equation of two independent
is a set of notes entitled "Aspect of time series," and it
discusses some aspects, mainly mathematical, of time series
analysis. It is written as a notes for a course on time
series to accompany in introductory textbook on the subject.
is a set of notes entitled "Cardinalities," an elementary
discussion of countably infinite and continuum cardinalities.
It also includes a discussion of the
Cantor-Schröder-Bernstein theorem saying that if
each of two sets can be mapped into the other in a one-to-one
way then the to sets can also be mapped onto each other in
a one-to-one way. The proof described in two different ways,
one is the standard way, the other is a different way with
is a short note describing the Euclidean algorithm to calculate
the greatest common divisor of two numbers and to represent
this greatest common divisor as a linear combination of the two
numbers. This is part of a larger set of notes
entitled "Introduction to open computing"
to be found at
The notes discuss Unix and Pascal on an elementary level.
These notes have not been updated for a while, and a
large part of it is now obsolete, so it can still be
used as an elementary introduction to using Unix or
discusses basic concepts of propositional and first order
predicate logic. It is part of the notes
is a short note entitled "Approximation of numbers by fractions"
gives an application of the pigeon-hole principle to show
that every real number can be approximated with a fraction
such that the goodness of the approximation is quadratic in
terms of the denominator of the fraction.
is a short note with the title "Irrationality of square roots"
showing that the square root of a positive integer is either
an integer or it is irrational. The proof does not rely
on the standard argument using divisibility by a prime
occurring with an odd exponent in the prime factorization
of the number, and it was apparently first described in
the twentieth century.
is a short note discussing some notable telescoping sums
involving products of consecutive integers or the
reciprocals of products of consecutive integers.
is a link to a writing called
"The greatest math mistake of the century," meaning the
twentieth century, is a humorous story involving Bertrand Russel,
Alfred North Whitehead,
David Hilbert, Kurt Gödel, Alan Turing, and John
von Neumann leading to the invention of the von Neumann
computer architecture, the standard architecture of modern
The author of the story is
unknown to me; it was posted for many years on the
Internet, then disappeared from one location and appeared
at another. I rescued the writing and I post it now
before it lost is forever. I welcome any information about
the origin of these notes.
contains a problem set with and without solutions as
problems.pdf and solutions.pdf related to some material
in the book Edward B. Burger and Michael Starbird,
The Heart of Mathematics: An Invitation to Effective
Thinking, Wiley, various editions, that I used in
a basic mathematics course.
Sun Dec 18 14:48:04 EST 2016.