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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 14 files changed, 358 insertions(+), 358 deletions(-) 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]_ - -.. list-table:: 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:`$a-b$` - - :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 - -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. McGraw-Hill, 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]_ + +.. list-table:: 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:`$a-b$` + - :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 + +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. McGraw-Hill, 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 1-4 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 1-4 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 1-4 - -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 1-4 + +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]_ -(1-12). - -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:`$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$`, 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 -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 -================== -*aleph-zero* (: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 *aleph-one* (:latex:`$\aleph_1$`). - -.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 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]_ +(1-12). + +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:`$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$`, 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 +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 +================== +*aleph-zero* (: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 *aleph-one* (:latex:`$\aleph_1$`). + +.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 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. McGraw-Hill, 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. McGraw-Hill, 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 rank-3 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:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$` - -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:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$` - -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 rank-3 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:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$` + +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:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$` + +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)=\{x|a