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-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.pageindex 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.pageindex 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 1-4diff --git a/math/logic.page b/math/logic.pageindex 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.pageindex 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.pageindex 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.pageindex 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.pageindex 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