math fixes

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.