Maths: Difference between revisions

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===Structures===
===Structures===
* '''Set''' - A collection of elements.
* '''Set''' - A collection of elements.
* '''Group''' - A set with a binary operation. The operation must be associative. If the operation is also commutative then it is an Abelian group. [TODO]: Get Axioms.
* '''Group''' - A set with a binary operation. The operation must be associative. If the operation is also commutative then it is an Abelian group. [https://www.youtube.com/watch?v=WwndchnEDS4 Bill Shillito]. [TODO]: Get Axioms.
* '''Field''' - A set with 2 binary operations (+, ×). Both must be commutative and associative. And have an identity elements (+ is 0 and × is 1). Every element must have an inverse (x = -x, x = 1/x). Must follow the distributive law. Must be closed under addition, multiplication, taking of inverses (Results must in in the same set).
* '''Field''' - A set with 2 binary operations (+, ×). Both must be commutative and associative. And have an identity elements (+ is 0 and × is 1). Every element must have an inverse (x = -x, x = 1/x). Must follow the distributive law. Must be closed under addition, multiplication, taking of inverses (Results must in in the same set).
* '''Vector Space''' - Like a 2D or 3D plane. Can be built from 'unit vectors' or '[https://www.youtube.com/watch?v=P2LTAUO1TdA basis]'. for example (1,0), (0,1). There was a Youtube on the topic. ℝ², ℝ³, ℝⁿ a vector space of n-dimensions over the field ℝ (ℝ is the scalar type).
* '''Vector Space''' - Like a 2D or 3D plane. Can be built from 'unit vectors' or '[https://www.youtube.com/watch?v=P2LTAUO1TdA basis]'. for example (1,0), (0,1). There was a Youtube on the topic. ℝ², ℝ³, ℝⁿ a vector space of n-dimensions over the field ℝ (ℝ is the scalar type).

Revision as of 05:17, 18 February 2017

Notation and Symbols

Operations

Wikipedia: Mathematical operators and symbols in Unicode Wikipedia: Supplemental Mathematical Operators

  • ⊕ - XOR. Direct Sum.
  • ⊖ - Symmetric difference.
  • ⊗ - Tensor product.
  • ⊘ -
  • ⊙ - Circled dot operator. ⨀ n-ary circled dot operator.
  • ∗ - Astrisk Operator. Also ✱.
  • ∘ - Function composition. Ring operator. (Also ∙ bullet operator)
  • ⋈ - Natural join.
  • ⋆ - Star Operator.

Calculus

  • ∫ ∬ ∭ ∮ ∯ ∰ - Integrals (double+triple). ∮ ∯ ∰ - Contour, surface and volume integrals. ∱ ∲ ∳ - Clockwise integral, clowise contour integral. [1]
  • ∆ - ∆x change in x. Sometimes symmetric difference.
  • ∇ - Gradient.

Other

  • ≠ ≈ ≡ ∃ ∄ ∈ ∉ ∌ ⇔ ∀ ∧ ∨ ¬ ≤ ≥ ≮ ≯ ⇒ ∩ ∪ ⋂ ⋃ ⊃ ⊂ ⊇ ⊆ ⊢
  • Arithmetic: − + ± × ⋅ · ÷ ⁄ √ ∛ ∜
  • x′ - x prime.
  • Misc: ∑ ℵ ℶ ∞ ∴ ∵ ! ∎ → ⟨ ⟩ ㏒ * ∥ ∦
  • 𝓋⃗, v⃗ - Vector 𝓋 + (U+20D7).
  • ∅ - empty set
  • ∂ - Partial Derivative.
  • δ - Kronecker Delta
  • ∏ - Product over (Like ∑)
  • ⊤ - True
  • ⊥ - False
  • ∖ - Set minus.
  • σ - Selection
  • {♠, ♦, ♥, ♣}


  • Subscript: ₀₁₂₃₄₅₆₇₈₉ ₊₋₌₍₎ ₐₑₒₓₔₕₖₗₘₙₚₛₜ
  • Superscript: ⁰¹²³⁴⁵⁶⁷⁸⁹ ⁺⁻⁼⁽⁾ ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖʳˢᵗᵘᵛʷˣʸᶻ
  • Blackboard Bold: 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ 𝕒𝕓𝕔𝕕𝕖𝕗𝕘𝕙𝕚𝕛𝕜𝕝𝕞𝕟𝕠𝕡𝕢𝕣𝕤𝕥𝕦𝕧𝕨𝕩𝕪𝕫 𝟘𝟙𝟚𝟛𝟜𝟝𝟞𝟟𝟠𝟡 ℾℽℿℼ⅀ ⅅⅆⅇⅈⅉ


  • wikipedia
  • Mathematical Italic: 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧


  • |𝐴|×|𝐵|

Glossary

Number Systems

  • ℕ - The set of all natural numbers. No zero. No negatives. Subtraction and division aren't always possible. ℕ = {1, 2, 3, ...}.
  • ℤ - The set of all integers. Subtraction is always possible. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
  • ℚ - The set of all rational numbers (Quotient). Fractions. Division is always possible. ℚ is a field.
  • ℝ - The set of all real numbers. Includes irrational and transcendental numbers. √(2), π, e, φ. ℝ is an extension of ℚ to a larger field.
  • ℝ⁺ - The set of all positive reals.
  • ℂ - The set of all complex numbers. i²=-1. Also a field.
  • ℍ - The set of all Quaternions.
  • 𝔽ₚ - Set of integers modulo a prime p. Addition and multiplication also defined modulo p.
  • ℚ(γ) - γ is the root of the polynomial x³-x-1. When γ³ appears it is replaced with γ+1. See: 2.2 in princeton companion of mathematics. AUTOMORPHISM
  • ℙ - Primes.

Algebraic Structures

Properties

  • associative - a(bc) = (ab)c
  • commutative - ab = ba
  • homomorphism - A category of function. Preserves the structure. f: X→Y. f(a)f(b)=f(c) and ab=c. A homomorphism of vector spaces is refered to as a linear map.
  • bijection - A bijection is invertable.
  • isomorphism - A homomorphism that is also a bijection. An operation-preserving bijection. A homomorphism with an inverse that is also a homomorphism. f: X→Y, g: Y→X
  • automorphism - Galios Groups related. An isomorphism from a structure to itself.
  • Abelion -

Structures

  • Set - A collection of elements.
  • Group - A set with a binary operation. The operation must be associative. If the operation is also commutative then it is an Abelian group. Bill Shillito. [TODO]: Get Axioms.
  • Field - A set with 2 binary operations (+, ×). Both must be commutative and associative. And have an identity elements (+ is 0 and × is 1). Every element must have an inverse (x = -x, x = 1/x). Must follow the distributive law. Must be closed under addition, multiplication, taking of inverses (Results must in in the same set).
  • Vector Space - Like a 2D or 3D plane. Can be built from 'unit vectors' or 'basis'. for example (1,0), (0,1). There was a Youtube on the topic. ℝ², ℝ³, ℝⁿ a vector space of n-dimensions over the field ℝ (ℝ is the scalar type).
  • Scalar - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
  • Ring - Similar to a field but multiplication doesn't require an inverse. Multiplication might not be commutative. + must be abelian. × must be closed and associative. Bill Shillito - Rings and Fields

Groups

  • Dihedral Groups: Dₙ - D₃ is the dihedral group of the triangle.
  • Symmetry Groups: Sₙ - S₃ the symmetric group of 3 elements. S₃ is isomorphic to D₃. (S₃ ≅ D₃)
  • Special Orthogonal: SO(n) - special orthogonal group (rotation group?).
  • General Linear: GL(n)

Eignvectors

  • Eigenvector - A vector in a vector space doesn't get rotated by a linear transformation. It stays on it's 'span'. It's the axis of rotation.
  • Eigenvalue - The value an Eigenvector is scaled by. 𝛢𝓋⃗=λ𝓋⃗. 𝛢 is transformation matrix. 𝓋⃗ is an Eigenvector. λ is the eigenvalue.


Analysis

Calculus

  • Derivatives - Amount of change.
  • Integral - Area under a function.
  • Partial derivative - For example, a 2D slice of a 3D surface.

Gradients

Videos

MOOCs


YouTube

Unsorted

HOW TO LEARN ADVANCED MATHEMATICS WITHOUT HEADING TO UNIVERSITY - PART 1 PART 3

4chan - Math Textbook Recommendations

Harvard Course of Abstract Algebra (apparently goes well with the Artin book)

List of books


Order of abstract algebra


This guide on algebra

This guide to imaginary numbers

Math intuition

These guides


These notes are recommended

HOW TO LEARN MATH: FOR STUDENTS

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