From f61026119df4700f69eb73e95620bc5928ca0fcb Mon Sep 17 00:00:00 2001
From: User
Date: Tue, 13 Oct 2009 02:52:09 +0000
Subject: Grand rename for gitit transfer

math/algebra  79 
math/algebra.page  79 ++++++++++++++++++++++++++++++++++++++++++++++++++++
math/integers  5 
math/integers.page  5 ++++
math/logic  18 
math/logic.page  18 ++++++++++++
math/numbers  54 
math/numbers.page  54 ++++++++++++++++++++++++++++++++++++
math/sets  47 
math/sets.page  47 +++++++++++++++++++++++++++++++
math/tensors  74 
math/tensors.page  74 +++++++++++++++++++++++++++++++++++++++++++++++++
math/topology  81 
math/topology.page  81 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
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delete mode 100644 math/algebra
create mode 100644 math/algebra.page
delete mode 100644 math/integers
create mode 100644 math/integers.page
delete mode 100644 math/logic
create mode 100644 math/logic.page
delete mode 100644 math/numbers
create mode 100644 math/numbers.page
delete mode 100644 math/sets
create mode 100644 math/sets.page
delete mode 100644 math/tensors
create mode 100644 math/tensors.page
delete mode 100644 math/topology
create mode 100644 math/topology.page
(limited to 'math')
diff git a/math/algebra b/math/algebra
deleted file mode 100644
index 3e39ddb..0000000
 a/math/algebra
+++ /dev/null
@@ 1,79 +0,0 @@
====================
Algebra
====================

.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_

.. listtable:: Closure of binary operators on given sets of numbers

 *  Operation name
  addition
  product
  subtraction
  division
  power
  root
 *  Operation symbol
  :latex:`$a + b$`
  :latex:`$a\times b$`
  :latex:`$ab$`
  :latex:`$\frac{a}{b}$`
  :latex:`$a^b$`
  :latex:`$\sqrt{\text{a}}$`
 *  Positive Integers
  Y
  Y
  N
  N
  Y
  N
 *  Positive rationals
  Y
  Y
  N
  Y
  Y
  N
 *  Rationals (and zero)
  Y
  Y
  Y
  Y
  Y
  N
 *  Reals wrt positive integers
  Y
  Y
  Y
  Y
  Y
  Y
 *  Complex numbers
  Y
  Y
  Y
  Y
  Y
  Y

Definitions
=============

*involution*
 to raise a number to a given power

*evolution*
 to take a given root of a number

*associative*
 :latex:`$(a+b)+c=a+(b+c)$`

*comutative*
 :latex:`$a+b=b+c$`

*distributive*
 :latex:`$(a+b)c=ac+bc$`

.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGrawHill, 1976

.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff git a/math/algebra.page b/math/algebra.page
new file mode 100644
index 0000000..3e39ddb
 /dev/null
+++ b/math/algebra.page
@@ 0,0 +1,79 @@
+====================
+Algebra
+====================
+
+.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+
+.. listtable:: Closure of binary operators on given sets of numbers
+
+ *  Operation name
+  addition
+  product
+  subtraction
+  division
+  power
+  root
+ *  Operation symbol
+  :latex:`$a + b$`
+  :latex:`$a\times b$`
+  :latex:`$ab$`
+  :latex:`$\frac{a}{b}$`
+  :latex:`$a^b$`
+  :latex:`$\sqrt{\text{a}}$`
+ *  Positive Integers
+  Y
+  Y
+  N
+  N
+  Y
+  N
+ *  Positive rationals
+  Y
+  Y
+  N
+  Y
+  Y
+  N
+ *  Rationals (and zero)
+  Y
+  Y
+  Y
+  Y
+  Y
+  N
+ *  Reals wrt positive integers
+  Y
+  Y
+  Y
+  Y
+  Y
+  Y
+ *  Complex numbers
+  Y
+  Y
+  Y
+  Y
+  Y
+  Y
+
+Definitions
+=============
+
+*involution*
+ to raise a number to a given power
+
+*evolution*
+ to take a given root of a number
+
+*associative*
+ :latex:`$(a+b)+c=a+(b+c)$`
+
+*comutative*
+ :latex:`$a+b=b+c$`
+
+*distributive*
+ :latex:`$(a+b)c=ac+bc$`
+
+.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGrawHill, 1976
+
+.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff git a/math/integers b/math/integers
deleted file mode 100644
index b946ce8..0000000
 a/math/integers
+++ /dev/null
@@ 1,5 +0,0 @@
====================
Integers
====================

definition via Peano's Postulates: meserev 14
diff git a/math/integers.page b/math/integers.page
new file mode 100644
index 0000000..b946ce8
 /dev/null
+++ b/math/integers.page
@@ 0,0 +1,5 @@
+====================
+Integers
+====================
+
+definition via Peano's Postulates: meserev 14
diff git a/math/logic b/math/logic
deleted file mode 100644
index 65ceba9..0000000
 a/math/logic
+++ /dev/null
@@ 1,18 +0,0 @@
=======================
Mathematical Logic
=======================

.. note::
 Incomplete; in progress

definition of induction: meserev 14

Proofs
===========

Indirect Proof: "reductio ad absurdum"
 Show a paradox or impossibility in all cases by assuming the proposition
 is false; then the proposition is true.

Proof by elimination
 Propose a complete set of propositions and remove all but one.
diff git a/math/logic.page b/math/logic.page
new file mode 100644
index 0000000..65ceba9
 /dev/null
+++ b/math/logic.page
@@ 0,0 +1,18 @@
+=======================
+Mathematical Logic
+=======================
+
+.. note::
+ Incomplete; in progress
+
+definition of induction: meserev 14
+
+Proofs
+===========
+
+Indirect Proof: "reductio ad absurdum"
+ Show a paradox or impossibility in all cases by assuming the proposition
+ is false; then the proposition is true.
+
+Proof by elimination
+ Propose a complete set of propositions and remove all but one.
diff git a/math/numbers b/math/numbers
deleted file mode 100644
index 541d174..0000000
 a/math/numbers
+++ /dev/null
@@ 1,54 +0,0 @@
========================
Numbers
========================

.. note::
 incomplete

.. note::
 Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_

.. contents::

*incommensurable*
 objects are incommensurable when their ratio isn't rational

Real Numbers
==================
The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_
(112).

Complex Numbers
==================
The *complex numbers* are constructed as an ordered pair of real numbers.

Algebraic and Transendental Numbers
===============================================
*Algebraic numbers* are solutions of polynomials, such as x in
:latex:`$a_0 x^n + a_1 x^{n1} + a_2 x^{n2} + ... a_n = 0$`, where all a are
real numbers. *Transcendental numbers* are not solutions to any such
polynomials.

All real numbers are either algebraic or transcendental.

Some algebraic numbers aren't real (such as :latex:`$i = \sqrt{1}$`). They
can be rational or irrational. All transcendental numbers are irrational;
some are not real.

Exersize: is the square root of 5 algebraic or transcendental?

e
========
:latex:`$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$`

Infinities
==================
*alephzero* (:latex:`$\aleph_0$`) is the countably infinite set.

Positive integers, integers, and rational numbers are all countably infinite.

It is unproven that the real numbers are *alephone* (:latex:`$\aleph_1$`).

.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGrawHill, 1976

.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff git a/math/numbers.page b/math/numbers.page
new file mode 100644
index 0000000..541d174
 /dev/null
+++ b/math/numbers.page
@@ 0,0 +1,54 @@
+========================
+Numbers
+========================
+
+.. note::
+ incomplete
+
+.. note::
+ Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+
+.. contents::
+
+*incommensurable*
+ objects are incommensurable when their ratio isn't rational
+
+Real Numbers
+==================
+The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_
+(112).
+
+Complex Numbers
+==================
+The *complex numbers* are constructed as an ordered pair of real numbers.
+
+Algebraic and Transendental Numbers
+===============================================
+*Algebraic numbers* are solutions of polynomials, such as x in
+:latex:`$a_0 x^n + a_1 x^{n1} + a_2 x^{n2} + ... a_n = 0$`, where all a are
+real numbers. *Transcendental numbers* are not solutions to any such
+polynomials.
+
+All real numbers are either algebraic or transcendental.
+
+Some algebraic numbers aren't real (such as :latex:`$i = \sqrt{1}$`). They
+can be rational or irrational. All transcendental numbers are irrational;
+some are not real.
+
+Exersize: is the square root of 5 algebraic or transcendental?
+
+e
+========
+:latex:`$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$`
+
+Infinities
+==================
+*alephzero* (:latex:`$\aleph_0$`) is the countably infinite set.
+
+Positive integers, integers, and rational numbers are all countably infinite.
+
+It is unproven that the real numbers are *alephone* (:latex:`$\aleph_1$`).
+
+.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGrawHill, 1976
+
+.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff git a/math/sets b/math/sets
deleted file mode 100644
index 42eb831..0000000
 a/math/sets
+++ /dev/null
@@ 1,47 +0,0 @@
====================
Sets
====================

.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_

Basics
=============
If every element :latex:`$a \in A$` is also :latex:`$a \in B$`, then we call
A a *subset* of B and write :latex:`$A \subset B$`. If there are elements of B
which are not elements of A, then we call A a *proper subset* of B.

If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`;
otherwise :latex:`$A \neq B$`.

The null or empty set, which has no elements, is a subset of all others.

A relation on a space of sets S is something that can be definted as either
true or false (holding or not holding) for any binary pair in S.

Binary Operators
==================
Binary operators defined on a set apply to any two elements of that set; order
may or may not be important. A set is *closed* with regards to a binary
operator if it contains the result of the binary operator. A set is *uniquely
defined* with regards to a binary operator if the result of the operator on two
elements of the set is unique from the results from all other pairs of
elements.

Some equivalence relations are
:latex:`$\identity$` (NOTE: = with three lines) (*identity*);
:latex:`$\congruence$` (NOTE: = with tilde on top) (*congruence*; eg of
geometric figures); and
:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures).

Some properties of equivalence relations are

*reflexive*
 if :latex:`$a=a$` is true for all a
*symetric*
 if :latex:`$a=b$` implies :latex:`$b=a$`
*transitive*
 if :latex:`$a=b$` and :latex:`$b=c$` implies :latex:`$a=c$`

.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGrawHill, 1976

.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff git a/math/sets.page b/math/sets.page
new file mode 100644
index 0000000..42eb831
 /dev/null
+++ b/math/sets.page
@@ 0,0 +1,47 @@
+====================
+Sets
+====================
+
+.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+
+Basics
+=============
+If every element :latex:`$a \in A$` is also :latex:`$a \in B$`, then we call
+A a *subset* of B and write :latex:`$A \subset B$`. If there are elements of B
+which are not elements of A, then we call A a *proper subset* of B.
+
+If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`;
+otherwise :latex:`$A \neq B$`.
+
+The null or empty set, which has no elements, is a subset of all others.
+
+A relation on a space of sets S is something that can be definted as either
+true or false (holding or not holding) for any binary pair in S.
+
+Binary Operators
+==================
+Binary operators defined on a set apply to any two elements of that set; order
+may or may not be important. A set is *closed* with regards to a binary
+operator if it contains the result of the binary operator. A set is *uniquely
+defined* with regards to a binary operator if the result of the operator on two
+elements of the set is unique from the results from all other pairs of
+elements.
+
+Some equivalence relations are
+:latex:`$\identity$` (NOTE: = with three lines) (*identity*);
+:latex:`$\congruence$` (NOTE: = with tilde on top) (*congruence*; eg of
+geometric figures); and
+:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures).
+
+Some properties of equivalence relations are
+
+*reflexive*
+ if :latex:`$a=a$` is true for all a
+*symetric*
+ if :latex:`$a=b$` implies :latex:`$b=a$`
+*transitive*
+ if :latex:`$a=b$` and :latex:`$b=c$` implies :latex:`$a=c$`
+
+.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGrawHill, 1976
+
+.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff git a/math/tensors b/math/tensors
deleted file mode 100644
index d46810e..0000000
 a/math/tensors
+++ /dev/null
@@ 1,74 +0,0 @@
============================================
Tensors, Differential Geometry, Manifolds
============================================

.. note:: Most of this content is based on a 2002 Caltech course taught by
 Kip Thorn [PH237]_


On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold.

There are points (:m:`$P$`), separation vectors (:m:`$\Delta \vector P$`),
curves (:m:`$Q(\zeta)$`), tangent vectors (:m:`$\delta P / \delta \zeta \equiv
\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) 
Q(\zeta) } }{\delta \zeta}$`)

Coordinates: :m:`$\Chi^\alpha (P)$`, where :m:`$\alpha = 0,1,2,3$`;
:m:`$Q(\Chi_0, \Chi_1, ...)$`
 there is an isomorphism between points and coordinates

Coordinate basis: :m:`$\vector{e_\alpha} \equiv \left( \frac{\partial
Q}{\partial \Chi^\alpha} \right$`)

 for instance, on a sphere with angles :m:`$\omega, \phi$`:

 :m:`$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$`

Components of a vector:

 :m:`$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$`

Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
 :m:`$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$`

Mathematicians like to say that the coordinate bases are actually directional derivatives

Tensors


A **tensor** :m:`$\bold{T}$` has a number of slots (called it's **rank**), takes a vector in each slot, and returns a real number. It is linear in vectors;
as an example for a rank3 tensor:

:m:`$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =
\alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \bold{T}
(\vector{B}, \vector{C}, \vector{D}) $$`

Even a regular vector is a tensor: pass it a second vector and take the
inner product (aka dot product) to get a real.

Define the **metric tensor**
:m:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$`. The
metric tensor is rank two and symetric (the vectors A and B could be swapped
without changing the scalar output value) and is the same as the inner product.

:m:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2  (AB)^2 ]$$`

Starting with individual vectors, we can construct tensors by taking the
product of their inner products with empty slots; for example

:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$`
:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$`

Spacetime


Two types of vectors.

Timelike: :m:`$\vector{\Delta P}$`
 :m:`$(\vector{\Delta P})^2 = (\Delta \Tau)^2$`

Spacelike: :m:`$\vector{\Delta Q}$`
 :m:`$(\vector{\Delta Q})^2 = +(\Delta S)^2$`

Because product of "up" and "down" basis vectors must be a positive Kronecker
delta, and timelikes squared come out negative, the time "up" basis must be negative of the time "down" basis vector.
diff git a/math/tensors.page b/math/tensors.page
new file mode 100644
index 0000000..d46810e
 /dev/null
+++ b/math/tensors.page
@@ 0,0 +1,74 @@
+============================================
+Tensors, Differential Geometry, Manifolds
+============================================
+
+.. note:: Most of this content is based on a 2002 Caltech course taught by
+ Kip Thorn [PH237]_
+
+
+On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold.
+
+There are points (:m:`$P$`), separation vectors (:m:`$\Delta \vector P$`),
+curves (:m:`$Q(\zeta)$`), tangent vectors (:m:`$\delta P / \delta \zeta \equiv
+\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) 
+Q(\zeta) } }{\delta \zeta}$`)
+
+Coordinates: :m:`$\Chi^\alpha (P)$`, where :m:`$\alpha = 0,1,2,3$`;
+:m:`$Q(\Chi_0, \Chi_1, ...)$`
+ there is an isomorphism between points and coordinates
+
+Coordinate basis: :m:`$\vector{e_\alpha} \equiv \left( \frac{\partial
+Q}{\partial \Chi^\alpha} \right$`)
+
+ for instance, on a sphere with angles :m:`$\omega, \phi$`:
+
+ :m:`$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$`
+
+Components of a vector:
+
+ :m:`$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$`
+
+Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
+ :m:`$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$`
+
+Mathematicians like to say that the coordinate bases are actually directional derivatives
+
+Tensors
+
+
+A **tensor** :m:`$\bold{T}$` has a number of slots (called it's **rank**), takes a vector in each slot, and returns a real number. It is linear in vectors;
+as an example for a rank3 tensor:
+
+:m:`$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =
+\alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \bold{T}
+(\vector{B}, \vector{C}, \vector{D}) $$`
+
+Even a regular vector is a tensor: pass it a second vector and take the
+inner product (aka dot product) to get a real.
+
+Define the **metric tensor**
+:m:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$`. The
+metric tensor is rank two and symetric (the vectors A and B could be swapped
+without changing the scalar output value) and is the same as the inner product.
+
+:m:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2  (AB)^2 ]$$`
+
+Starting with individual vectors, we can construct tensors by taking the
+product of their inner products with empty slots; for example
+
+:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$`
+:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$`
+
+Spacetime
+
+
+Two types of vectors.
+
+Timelike: :m:`$\vector{\Delta P}$`
+ :m:`$(\vector{\Delta P})^2 = (\Delta \Tau)^2$`
+
+Spacelike: :m:`$\vector{\Delta Q}$`
+ :m:`$(\vector{\Delta Q})^2 = +(\Delta S)^2$`
+
+Because product of "up" and "down" basis vectors must be a positive Kronecker
+delta, and timelikes squared come out negative, the time "up" basis must be negative of the time "down" basis vector.
diff git a/math/topology b/math/topology
deleted file mode 100644
index 6f03eee..0000000
 a/math/topology
+++ /dev/null
@@ 1,81 +0,0 @@
====================
Topology
====================

.. warning:: Incomplete; in progress

.. note:: Most of the definitions and notation in the section are based on [munkres]_

A *topological space* is a set for which a valid topology has been defined: the topology
determines which subsets of the topological space are open and closed. In this way the
concept of open and closed subsets on the real number line (such as :m:`$(0,1)$` and
:m:`$[1,2]$`) are generalized to arbitrary sets.

Formally, a *topology* on a set :m:`$A$` is a collection :m:`$\mathcal{T}$` of
subsets of :m:`$A$` fufiling the criteria:

 1. The empty set and the entire set :m:`$A$` are both in :m:`$\mathcal{T}$`.

 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is
 also in :m:`$\mathcal{T}$`.

 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is
 also in :m:`$\mathcal{T}$`.

If a subset :m:`$B$` of :m:`$A$` is a member of :m:`$\mathcal{T}$` then
:m:`$B$` is an open set under the topology :m:`$\mathcal{T}$`.

*Coarseness* and *Fineness* are ways of comparing two topologies on the same space.
:m:`$\mathcal{T'}$` is finer than :m:`$\mathcal{T}$` if :m:`$\mathcal{T}$`
is a subset of :m:`$\mathcal{T'}$` (and :m:`$\mathcal{T}$` is coarser);
it is *strictly finer* if it is a proper subset (and :m:`$\mathcal{T}$` is
*strictly coarser*). Two sets are *comprable* if either :m:`$\mathcal{T\in T'}$`
or :m:`$\mathcal{T'\in T}$`.
*Smaller* and *larger* are somtimes used instead of finer and coarser.

Topologies can be generated from a *basis*.

TODO: Hausdorf

Frequently Used Topologies
============================

*Standard Topology*
 The standard topology on the real line is generated by the collection of all intervals
 :m:`$$(a,b)=\{xa