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-rw-r--r-- | math/algebra.page | 91 | ||||

-rw-r--r-- | math/integers.page | 11 | ||||

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-rw-r--r-- | math/numbers.page | 54 | ||||

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-rw-r--r-- | math/topology.page | 73 |

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. |