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authorbnewbold <bnewbold@ziggy.(none)>2010-01-24 03:39:25 -0500
committerbnewbold <bnewbold@ziggy.(none)>2010-01-24 03:39:25 -0500
commita290d583ca3c4dfc39115068f209d64449c93a03 (patch)
tree558d8a8905d076295c17cbc013efde39a76fe259 /math
parent88763d7db3f803b9e5b6351e01c186a98e50bbf2 (diff)
downloadknowledge-a290d583ca3c4dfc39115068f209d64449c93a03.tar.gz
knowledge-a290d583ca3c4dfc39115068f209d64449c93a03.zip
math fixes
Diffstat (limited to 'math')
-rw-r--r--math/algebra.page91
-rw-r--r--math/integers.page11
-rw-r--r--math/logic.page16
-rw-r--r--math/numbers.page54
-rw-r--r--math/sets.page51
-rw-r--r--math/tensors.page44
-rw-r--r--math/topology.page73
7 files changed, 151 insertions, 189 deletions
diff --git a/math/algebra.page b/math/algebra.page
index e1a44db..658267e 100644
--- a/math/algebra.page
+++ b/math/algebra.page
@@ -1,85 +1,42 @@
---
-format: rst
+format: markdown
categories: math
toc: no
...
-====================
-Algebra
-====================
+# Algebra
-.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+*Note: Most of the definitions and notation in the section are based on [rudin] or [meserve].*
-.. list-table:: Closure of binary operators on given sets of numbers
+Name Symbol Pos. Integers? Pos. Rationals? Rationals? Reals (wrt Pos Int.)? Complex?
+---- ----------------- -------------- --------------- ---------- --------------------- --------
+addition $a + b$ Y Y Y Y Y
+product $a\times b$ Y Y Y Y Y
+subtraction $a-b$ N N Y Y Y
+division $\frac{a}{b}$ N Y Y Y Y
+power $a^b$ Y Y Y Y Y
+root $\sqrt{\text{a}}$ N N N Y Y
+---- ----------------- -------------- --------------- ---------- --------------------- --------
- * - Operation name
- - addition
- - product
- - subtraction
- - division
- - power
- - root
- * - Operation symbol
- - :latex:`$a + b$`
- - :latex:`$a\times b$`
- - :latex:`$a-b$`
- - :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
+Table: Closure of binary operators on given sets of numbers
-Definitions
-=============
+## Definitions
-*involution*
+involution
to raise a number to a given power
-*evolution*
+evolution
to take a given root of a number
-*associative*
- :latex:`$(a+b)+c=a+(b+c)$`
+associative
+ $(a+b)+c=a+(b+c)$
-*comutative*
- :latex:`$a+b=b+c$`
+comutative
+ $a+b=b+c$
-*distributive*
- :latex:`$(a+b)c=ac+bc$`
+distributive
+ $(a+b)c=ac+bc$
-.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976
+[^rudin] **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976
-.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
+[^meserve] **Fundamental Concepts of Algebra**, by Bruce Meserve.
diff --git a/math/integers.page b/math/integers.page
index b946ce8..e9a657c 100644
--- a/math/integers.page
+++ b/math/integers.page
@@ -1,5 +1,10 @@
-====================
-Integers
-====================
+---
+format: markdown
+categories: math
+toc: no
+...
+
+
+# Integers
definition via Peano's Postulates: meserev 1-4
diff --git a/math/logic.page b/math/logic.page
index 65ceba9..9d71823 100644
--- a/math/logic.page
+++ b/math/logic.page
@@ -1,14 +1,16 @@
-=======================
-Mathematical Logic
-=======================
+---
+format: markdown
+categories: math
+toc: no
+...
-.. note::
- Incomplete; in progress
+# Mathematical Logic
+
+*Note: Incomplete; in progress*
definition of induction: meserev 1-4
-Proofs
-===========
+## Proofs
Indirect Proof: "reductio ad absurdum"
Show a paradox or impossibility in all cases by assuming the proposition
diff --git a/math/numbers.page b/math/numbers.page
index 541d174..6481c75 100644
--- a/math/numbers.page
+++ b/math/numbers.page
@@ -1,54 +1,52 @@
-========================
-Numbers
-========================
+---
+format: markdown
+categories: math
+toc: no
+...
-.. note::
- incomplete
+# Numbers
-.. note::
- Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
-.. contents::
+*References: most of the definitions and notation in the section are based on [rudin] or [meserve]*
-*incommensurable*
+incommensurable
objects are incommensurable when their ratio isn't rational
-Real Numbers
-==================
-The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_
-(1-12).
+## Real Numbers
+
+The *real numbers* are defined via Dedakind cuts in [^rudin], or [^meserve]
+(p1-12).
+
+## Complex Numbers
-Complex Numbers
-==================
The *complex numbers* are constructed as an ordered pair of real numbers.
-Algebraic and Transendental Numbers
-===============================================
+## Algebraic and Transendental Numbers
+
*Algebraic numbers* are solutions of polynomials, such as x in
-:latex:`$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$`, where all a are
+$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$, 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
+Some algebraic numbers aren't real (such as $i = \sqrt{-1}$). 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:`$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$`
+## e
+$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$
+
+## Infinities
-Infinities
-==================
-*aleph-zero* (:latex:`$\aleph_0$`) is the countably infinite set.
+*aleph-zero* ($\aleph_0$) is the countably infinite set.
Positive integers, integers, and rational numbers are all countably infinite.
-It is unproven that the real numbers are *aleph-one* (:latex:`$\aleph_1$`).
+It is unproven that the real numbers are *aleph-one* ($\aleph_1$).
-.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976
+[^rudin] **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976
-.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
+[^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.
diff --git a/math/sets.page b/math/sets.page
index 42eb831..7b464ed 100644
--- a/math/sets.page
+++ b/math/sets.page
@@ -1,25 +1,30 @@
-====================
-Sets
-====================
+---
+format: markdown
+categories: math
+toc: no
+...
-.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+# Sets
-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
+*References: Most of the definitions and notation in the section are based on
+[^rudin] or [^meserve]*
+
+## Basics
+
+If every element $a \in A$ is also $a \in B$, then we call
+A a *subset* of B and write $A \subset B$. 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:`$A \neq B$`.
+If $A \supset B$ and $B \supset A$ we write $A = B$;
+otherwise $A \neq B$.
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
+
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
@@ -28,20 +33,20 @@ 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
+$\identity$ (NOTE: = with three lines) (*identity*);
+$\congruence$ (NOTE: = with tilde on top) (*congruence*; eg of
geometric figures); and
-:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures).
+$~$ (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:`$a=c$`
+reflexive
+ if $a=a$ is true for all a
+symetric
+ if $a=b$ implies $b=a$
+transitive
+ if $a=b$ and $b=c$ implies $a=c$
-.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976
+[^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976
-.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
+[^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.
diff --git a/math/tensors.page b/math/tensors.page
index e1d24fc..7ea1848 100644
--- a/math/tensors.page
+++ b/math/tensors.page
@@ -1,5 +1,3 @@
-
-
Tensors, Differential Geometry, Manifolds
============================================
@@ -9,66 +7,66 @@ Most of this content is based on a 2002 Caltech course taught by Kip Thorn (PH23
On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold.
There are points ($P$), separation vectors ($\Delta \vector P$),
-curves (:m:`$Q(\zeta)$`), tangent vectors (:m:`$\delta P / \delta \zeta \equiv
+curves ($Q(\zeta)$), tangent vectors ($\delta P / \delta \zeta \equiv
\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) -
-Q(\zeta) } }{\delta \zeta}$`)
+Q(\zeta) } }{\delta \zeta}$)
-Coordinates: :m:`$\Chi^\alpha (P)$`, where :m:`$\alpha = 0,1,2,3$`;
-:m:`$Q(\Chi_0, \Chi_1, ...)$`
+Coordinates: $\Chi^\alpha (P)$, where $\alpha = 0,1,2,3$;
+$Q(\Chi_0, \Chi_1, ...)$
there is an isomorphism between points and coordinates
-Coordinate basis: :m:`$\vector{e_\alpha} \equiv \left( \frac{\partial
-Q}{\partial \Chi^\alpha} \right$`)
+Coordinate basis: $\vector{e_\alpha} \equiv \left( \frac{\partial
+Q}{\partial \Chi^\alpha} \right$)
- for instance, on a sphere with angles :m:`$\omega, \phi$`:
+ for instance, on a sphere with angles $\omega, \phi$:
- :m:`$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$`
+ $\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$
Components of a vector:
- :m:`$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$`
+ $\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$
Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
- :m:`$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$`
+ $\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$
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;
+A **tensor** $\bold{T}$ 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 rank-3 tensor:
-:m:`$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =
+$$\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}) $$`
+(\vector{B}, \vector{C}, \vector{D}) $$
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
+$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$. 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:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$`
+$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$
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:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$`
+$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$
+$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$
Spacetime
--------------
Two types of vectors.
-Timelike: :m:`$\vector{\Delta P}$`
- :m:`$(\vector{\Delta P})^2 = -(\Delta \Tau)^2$`
+Timelike: $\vector{\Delta P}$
+ $(\vector{\Delta P})^2 = -(\Delta \Tau)^2$
-Spacelike: :m:`$\vector{\Delta Q}$`
- :m:`$(\vector{\Delta Q})^2 = +(\Delta S)^2$`
+Spacelike: $\vector{\Delta Q}$
+ $(\vector{\Delta Q})^2 = +(\Delta S)^2$
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.page b/math/topology.page
index 6f03eee..ea369fb 100644
--- a/math/topology.page
+++ b/math/topology.page
@@ -1,36 +1,33 @@
-====================
Topology
====================
-.. warning:: Incomplete; in progress
-
-.. note:: Most of the definitions and notation in the section are based on [munkres]_
+*References: 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.
+concept of open and closed subsets on the real number line (such as $(0,1)$ and
+$[1,2]$) 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:
+Formally, a *topology* on a set $A$ is a collection $\mathcal{T}$ of
+subsets of $A$ fufiling the criteria:
- 1. The empty set and the entire set :m:`$A$` are both in :m:`$\mathcal{T}$`.
+ 1. The empty set and the entire set $A$ are both in $\mathcal{T}$.
- 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is
- also in :m:`$\mathcal{T}$`.
+ 2. The union of an arbitrary number of elements of $\mathcal{T}$ is
+ also in $\mathcal{T}$.
- 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is
- also in :m:`$\mathcal{T}$`.
+ 3. The intersection of a finite number of elements of $\mathcal{T}$ is
+ also in $\mathcal{T}$.
-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:`$\mathcal{T}$`.
+If a subset $B$ of $A$ is a member of $\mathcal{T}$ then
+$B$ is an open set under the topology $\mathcal{T}$.
*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:`$\mathcal{T'\in T}$`.
+$\mathcal{T'}$ is finer than $\mathcal{T}$ if $\mathcal{T}$
+is a subset of $\mathcal{T'}$ (and $\mathcal{T}$ is coarser);
+it is *strictly finer* if it is a proper subset (and $\mathcal{T}$ is
+*strictly coarser*). Two sets are *comprable* if either $\mathcal{T\in T'}$
+or $\mathcal{T'\in T}$.
*Smaller* and *larger* are somtimes used instead of finer and coarser.
Topologies can be generated from a *basis*.
@@ -42,40 +39,40 @@ Frequently Used Topologies
*Standard Topology*
The standard topology on the real line is generated by the collection of all intervals
- :m:`$$(a,b)=\{x|a<x<b\}$$`
+ $$(a,b)=\{x|a<x<b\}$$
This is the usual definition for open sets on the real line.
*Discrete Topology*
- The topology on a set :m:`$A$` consisting of all points of :m:`$A$`;
- in other words the power set of :m:`$A$`.
+ The topology on a set $A$ consisting of all points of $A$;
+ in other words the power set of $A$.
*Trivial/Indiscrete Topology*
- The topology on a set :m:`$A$` consisting of only the empty set and :m:`$A$`
- itself. Not super interesting but it's always there (when :m:`$A$` isn't empty).
+ The topology on a set $A$ consisting of only the empty set and $A$
+ itself. Not super interesting but it's always there (when $A$ isn't empty).
-*Finite Complement Topology* (:m:`$\mathcal{T}_f$`)
- The topology on a set :m:`$A$` consisting of the empty set any subset
- :m:`$U$` such that :m:`$A-U$` has a finite number of elements.
+*Finite Complement Topology* ($\mathcal{T}_f$)
+ The topology on a set $A$ consisting of the empty set any subset
+ $U$ such that $A-U$ has a finite number of elements.
-*Lower Limit Topology* (:m:`$\mathbb{R}_{\mathcal{l}}$`)
+*Lower Limit Topology* ($\mathbb{R}_{\mathcal{l}}$)
The lower limit topology on the real line is generated by the collection of all half open
intervals
- :m:`$$[a,b)=\{x|a\leq x<b\}$$`
- :m:`$\mathbb{R}_{\mathcal{l}}$` is strictly finer than the standard topology and
- is not comprable to :m:`$\mathbb{R}_K$`.
+ $$[a,b)=\{x|a\leq x<b\}$$
+ $\mathbb{R}_{\mathcal{l}}$ is strictly finer than the standard topology and
+ is not comprable to $\mathbb{R}_K$.
-*K-Topology* (:m:`$\mathbb{R}_K$`)
- Let :m:`$K$` denote the set of all numbers :m:`$1/n$` where :m:`$n$` is
+*K-Topology* ($\mathbb{R}_K$)
+ Let $K$ denote the set of all numbers $1/n$ where $n$ is
a positive integer.
The K-topology on the real line is generated by the collection of all standard open intervals
- minus :m:`$K$`.
- :m:`$\mathbb{R}_K$` is strictly finer than the standard topology and is not
- comprable to :m:`$\mathbb{R}_{\mathcal{l}}$`.
+ minus $K$.
+ $\mathbb{R}_K$ is strictly finer than the standard topology and is not
+ comprable to $\mathbb{R}_{\mathcal{l}}$.
*Order Topology*
TODO
-.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.
+[^munkres] **Topology (2nd edition)**, by James R. Munkres.