Part II

Again, for variable t, think

"There is a number; call it t".

That approach to the meaning of a
variable works well enough for
essentially all of math.

Calculus was invented mostly by I. Newton,
mostly for physics, especially for
explaining the motions of the planets.
Since then calculus has become a pillar of
civilization, especially Western
Civilization.

Mostly in calculus, integration is finding
the area (or more carefully defining the
area) under a curve, maybe a parabola.

But can also have a line integral, say,
the work do when carrying 100 pounds of
hay to the top of the hay loft of the
barn. Here we integrate from beginning to
end of the work. If we let the hay fall,
then neglecting friction, etc. get the
work back as energy. So, have to pay
attention to the direction of the
integration, from the ground up to the hay
loft or from the hay loft back to the
ground. The two values have opposite
signs.

You can teach yourself calculus: Just get
1-4, at least one good calculus texts and
dig in -- read the chapters, follow the
material, work nearly all the more
difficult exercises, and maybe check 1-4
times a year with a good mathematician. I
did that for freshman college calculus
(but didn't check with anyone). I never
took the course and, instead, started on
sophomore calculus and made As. No
problem.

(3) Linear Algebra. Likely the next
course would be linear algebra. It would
be good to do this subject 2-4 times from
more elementary treatments to some of the
more advanced material. Some of the more
advanced material can cover linear
programming optimization, classic two
person game theory, the proof of the Nash
result in game theory, some treatments of
the fast Fourier transform in digital
signal processing, group representations
in the quantum mechanics of molecular
spectroscopy, error correcting codes in
algebraic coding theory, numerical
methods, and more.

The start of linear algebra is just one or
several linear equations in one or
several variables. E.g., for two
equations in three variables, unknowns:

2x + 3y - 6z = 0

5x - 2y + 7z = 3

For positive integers (whole numbers) m
and n, given m such linear equations in
n variables (unknowns), the number of
solutions, depending also on the numerical
constants in the equations, is none, one,
or infinitely many.

The main way to find all the solutions is
just Gauss elimination.

That covers a good chunk of a first course
in linear algebra.

The notation

2x + 3y - 6z = 0

5x - 2y + 7z = 3

gets to be clumsy so we write instead

2 + 3 - 6

5 - 2 + 7

write that with big, square brackets, and
call it a matrix with two rows and three
columns. The individual numbers are
components. Of course, a matrix is
conceptually close to an array with two
subscripts common in programming
languages. Maybe we call the matrix

2 + 3 - 6

5 - 2 + 7

A. Since matrix A has 2 rows and 3
columns, we say that it is 2 x 3 ("2 by
3").

Then we write the x, y, z in a column

x

y

z

apply the big, square brackets, and call
it a matrix with 3 rows and 1 column and
maybe call it v. We write the right side

0

3

also with brackets, and maybe call it b.

Then we define a product, a matrix
product Av so that

Av = b

means just the same as

2x + 3y - 6z = 0

5x - 2y + 7z = 3

From then on we work with matrices and try
to avoid notation like

2x + 3y - 6z = 0

5x - 2y + 7z = 3

In this way, a course in linear algebra
is commonly called a course, Linear
Algebra and Matrix Theory.

A matrix with just one column is a column
vector; with just one row, a row
vector. In short, either is called a
vector.

Linearity: Given m x n matrices A and B,
m x 1 matrices u, v, and real numbers c,
d, we have that

A(cu + dv) = cAu + dAv

Here we regard Au as case of function A
with argument u. That is, we could have
said that matrix A defines function f so
that

f(u) = Au

With

cA + dA

we form products cA and dA and sums

cA + dA

So, we have to define both of those: The
definitions are close to just obvious.

The equation

A(cu + dv) = cAu + dAv

means that matrix A acts like a linear
function. Big deal! The real world is
awash in linear behavior, and linearity is
an enormously powerful property
mathematically. The rest of linear
algebra and matrix theory is nearly all
about the consequences of such linearity.

The fundamental theorem of algebra is that
each (high school style) polynomial in
variable x can be uniquely factored into a
product of terms of the form (ax + b)
where the a and b are possibly complex
numbers. Essentially from this fact, the
more advanced parts of linear algebra use
the complex numbers and not just the real
numbers.

At the end of a second course in linear
algebra, will pay attention to the
essentially geometrical notions of length
and angle and, in particular, to
orthogonality, that is, perpendicular.
Big topics are the Gram-Schmidt process
where find some orthogonal vectors and the
polar decomposition.

The polar decomposition result says that a
matrix acting on a circle will yield an
ellipse, that is, for matrix A and vector
u, if we let u take on the values of all
the points on a circle, then the Au will
generate all the values of all the points
on an ellipse. The ellipse has two axes,
and they are mutually perpendicular. So,
if have the vectors of the two axes, then
have enough to construct the whole
ellipse.

In linear algebra, this situation
generalizes to any finite dimension and
yields the singular value decomposition,
principle components, factor analysis, and
more.

With more advanced work, this situation
becomes the spectral theory of
self-adjoint linear operatiors central to
quantum mechanics.

(4) The Four Main Parts of Math

There are four main parts of math:

(A) Foundations, that is, deep in the
basement with set theory.

(B) Algebra, as in high school, the
integers, prime numbers, the fundamental
theorem of arithmetic, greatest common
divisor and least common multiple, the
fundamental theorem of algebra, linear
algebra and matrix theory, the
generalizations of number systems such as
groups, rings, fields (the rational, real,
and complex numbers are all fields but
so is the set of integers modulo a prime
number), etc. Fermat's last theorem,
settled by A. Wiles, is part of algebra;
so are the deep, difficult questions about
prime numbers, etc. There are
applications in error correcting codes and
cryptography.

(C) Geometry, e.g., high school plane
geometry, analytic geometry, and
differential geometry, a deep subject
important for relativity.

(D) Analysis, as in calculus, differential
equations, functional analysis, e.g.,
Banach and Hilbert spaces, partial
differential equations, e.g., as in
Maxwell's equations, fluid flow, and
relativity, probability, statistics,
stochastic processes, optimization, and
more.

In differential equations we are given an
equation with the derivatives of some
function and want to find the function.

E.g., for real valued function y(t) and
constants k and b, we might have

y'(t) = k y(t) (b - y(t))

where y'(t) = d/dt y(t), the first derivative of function y(t). At one time, that little differential
equation saved FedEx from going out of
business. The solution is a lazy S
curve and a first cut at viral growth.

There is topology which is partly in
geometry and analysis. The main idea of
topology is continuity, that is,
changing without sudden jumps or some
cases of wildly fast oscillations.

There is algebraic geometry partly in
algebra and geometry.

Much of number theory has deep connections
with analysis.

Much of linear algebra is an introduction
to functional analysis in analysis.

The older applied math is mostly analysis.

(5) Analysis. Usually after linear
algebra will study advanced calculus and
analysis. Broadly there are two
approaches, (A) theory and (B)
applications.

The theory is mostly to give fully careful
proofs of the results and first
generalizations of what you saw in
calculus. For the theory maybe the most
respected text is

R. Rudin, Principles of Mathematical
Analysis, Third Edition.

For this, get the third edition and not
either of the two earlier editions.

So, will discover that the integral
learned in calculus is called the Riemann
integral because B. Riemann made the
theorems solid. Rudin also does the easy
generalization to the Riemann-Stieltjes
integral.