Notation and Symbols

Operations

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

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

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

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.

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.
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

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)

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. v is Eigenvector. λ is the eigenvalue.