Searchable Index

Every equation, formula and topic across grades 9–12. Try quadratic, pH, or derivative.

• topic

  Elementary K–2 — Early Math

  ∑ Mathematics · Elementary K–2 — Early Math

• note

  Counting, number sense, place value to 100

  ∑ Mathematics · Elementary K–2 — Early Math

• note

  Addition & subtraction within 20

  ∑ Mathematics · Elementary K–2 — Early Math

• note

  Shapes, measurement, time, money basics

  ∑ Mathematics · Elementary K–2 — Early Math

• note

  Simple word problems

  ∑ Mathematics · Elementary K–2 — Early Math

• topic

  Elementary 3–5 — Arithmetic & Fractions

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• note

  Multiplication & division fluency

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• note

  Fractions, decimals, mixed numbers

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• note

  Area, perimeter, volume of rectangles

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• note

  Intro to coordinate plane & data graphs

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• equation

  Area rect: A = l * w

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• equation

  Perimeter: P = 2(l + w)

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• equation

  Volume box: V = l * w * h

  ∑ Mathematics · Elementary 3–5 — Arithmetic & Fractions

• topic

  Middle School 6–8 — Pre-Algebra

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• note

  Ratios, proportions, percents

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• note

  Integers, rational numbers, absolute value

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• note

  Expressions, one- and two-step equations

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• note

  Pythagorean theorem; intro to functions

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• equation

  Percent: part = percent * whole

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• equation

  Slope: m = (y2 - y1) / (x2 - x1)

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• equation

  Pythag.: a^2 + b^2 = c^2

  ∑ Mathematics · Middle School 6–8 — Pre-Algebra

• topic

  Grade 9 — Algebra I

  ∑ Mathematics · Grade 9 — Algebra I

• note

  Variables, expressions, order of operations (PEMDAS)

  ∑ Mathematics · Grade 9 — Algebra I

• note

  Linear equations & inequalities; systems

  ∑ Mathematics · Grade 9 — Algebra I

• note

  Functions, slope, lines

  ∑ Mathematics · Grade 9 — Algebra I

• note

  Exponents, polynomials, factoring

  ∑ Mathematics · Grade 9 — Algebra I

• note

  Quadratic equations

  ∑ Mathematics · Grade 9 — Algebra I

• equation

  Slope: m = (y2 - y1) / (x2 - x1)

  ∑ Mathematics · Grade 9 — Algebra I

• equation

  Slope-intercept: y = mx + b

  ∑ Mathematics · Grade 9 — Algebra I

• equation

  Quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / 2a

  ∑ Mathematics · Grade 9 — Algebra I

• equation

  Exponent rules: a^m * a^n = a^(m+n); (a^m)^n = a^(mn)

  ∑ Mathematics · Grade 9 — Algebra I

• topic

  Grade 10 — Geometry

  ∑ Mathematics · Grade 10 — Geometry

• note

  Lines, angles, parallel & perpendicular

  ∑ Mathematics · Grade 10 — Geometry

• note

  Triangle congruence & similarity

  ∑ Mathematics · Grade 10 — Geometry

• note

  Right-triangle trig (SOH-CAH-TOA)

  ∑ Mathematics · Grade 10 — Geometry

• note

  Circles, area, volume

  ∑ Mathematics · Grade 10 — Geometry

• note

  Two-column proofs

  ∑ Mathematics · Grade 10 — Geometry

• equation

  Pythagorean: a^2 + b^2 = c^2

  ∑ Mathematics · Grade 10 — Geometry

• equation

  Distance: d = sqrt((x2-x1)^2 + (y2-y1)^2)

  ∑ Mathematics · Grade 10 — Geometry

• equation

  Circle: (x-h)^2 + (y-k)^2 = r^2

  ∑ Mathematics · Grade 10 — Geometry

• equation

  Sphere: V = (4/3) pi r^3; SA = 4 pi r^2

  ∑ Mathematics · Grade 10 — Geometry

• equation

  Law of Cosines: c^2 = a^2 + b^2 - 2ab cos C

  ∑ Mathematics · Grade 10 — Geometry

• topic

  Grade 11 — Algebra II

  ∑ Mathematics · Grade 11 — Algebra II

• note

  Polynomial, rational, exponential, log functions

  ∑ Mathematics · Grade 11 — Algebra II

• note

  Complex numbers, conics

  ∑ Mathematics · Grade 11 — Algebra II

• note

  Sequences & series

  ∑ Mathematics · Grade 11 — Algebra II

• note

  Probability & statistics

  ∑ Mathematics · Grade 11 — Algebra II

• note

  Trig functions & identities

  ∑ Mathematics · Grade 11 — Algebra II

• equation

  Vertex form: y = a(x - h)^2 + k

  ∑ Mathematics · Grade 11 — Algebra II

• equation

  Log rules: log(xy) = log x + log y

  ∑ Mathematics · Grade 11 — Algebra II

• equation

  Exp growth: A = A0 * e^(kt)

  ∑ Mathematics · Grade 11 — Algebra II

• equation

  Compound int.: A = P(1 + r/n)^(nt)

  ∑ Mathematics · Grade 11 — Algebra II

• equation

  Geometric sum: Sn = a1 (1 - r^n) / (1 - r)

  ∑ Mathematics · Grade 11 — Algebra II

• equation

  Pythag identity: sin^2 + cos^2 = 1

  ∑ Mathematics · Grade 11 — Algebra II

• topic

  Grade 12 — Pre-Calc / Calculus

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• note

  Trig identities, polar, vectors

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• note

  Limits and continuity

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• note

  Derivatives & rules

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• note

  Optimization, related rates

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• note

  Integrals, FTC

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• equation

  Derivative: f'(x) = lim_{h->0} (f(x+h) - f(x))/h

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• equation

  Power rule: d/dx [x^n] = n x^(n-1)

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• equation

  Product rule: (fg)' = f'g + fg'

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• equation

  Chain rule: d/dx f(g(x)) = f'(g(x)) g'(x)

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• equation

  Integral: int x^n dx = x^(n+1)/(n+1) + C

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• equation

  FTC: int_a^b f(x) dx = F(b) - F(a)

  ∑ Mathematics · Grade 12 — Pre-Calc / Calculus

• topic

  College Freshman — Calculus I & II

  ∑ Mathematics · College Freshman — Calculus I & II

• note

  Limits, derivatives, applications

  ∑ Mathematics · College Freshman — Calculus I & II

• note

  Definite & indefinite integrals

  ∑ Mathematics · College Freshman — Calculus I & II

• note

  Techniques of integration, series

  ∑ Mathematics · College Freshman — Calculus I & II

• note

  Intro to differential equations

  ∑ Mathematics · College Freshman — Calculus I & II

• equation

  Taylor: f(x) = Σ f^(n)(a) (x-a)^n / n!

  ∑ Mathematics · College Freshman — Calculus I & II

• equation

  Integration by parts: ∫ u dv = uv - ∫ v du

  ∑ Mathematics · College Freshman — Calculus I & II

• topic

  College Sophomore — Multivariable & Linear Algebra

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• note

  Partial derivatives, gradient, multiple integrals

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• note

  Vector calculus: divergence, curl, Stokes

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• note

  Matrices, determinants, eigenvalues

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• note

  Vector spaces, linear transformations

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• equation

  Gradient: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• equation

  Det 2x2: det = ad - bc

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• equation

  Eigen: A v = λ v

  ∑ Mathematics · College Sophomore — Multivariable & Linear Algebra

• topic

  College Junior — Probability & Statistics

  ∑ Mathematics · College Junior — Probability & Statistics

• note

  Random variables, distributions

  ∑ Mathematics · College Junior — Probability & Statistics

• note

  Expectation, variance, CLT

  ∑ Mathematics · College Junior — Probability & Statistics

• note

  Hypothesis testing, confidence intervals

  ∑ Mathematics · College Junior — Probability & Statistics

• note

  Regression & ANOVA

  ∑ Mathematics · College Junior — Probability & Statistics

• equation

  Expectation: E[X] = Σ x p(x)

  ∑ Mathematics · College Junior — Probability & Statistics

• equation

  Variance: Var(X) = E[X^2] - E[X]^2

  ∑ Mathematics · College Junior — Probability & Statistics

• equation

  Z-score: z = (x - μ) / σ

  ∑ Mathematics · College Junior — Probability & Statistics

• topic

  College Senior — Real & Abstract Analysis

  ∑ Mathematics · College Senior — Real & Abstract Analysis

• note

  Real analysis: limits, continuity, measure

  ∑ Mathematics · College Senior — Real & Abstract Analysis

• note

  Abstract algebra: groups, rings, fields

  ∑ Mathematics · College Senior — Real & Abstract Analysis

• note

  Topology basics

  ∑ Mathematics · College Senior — Real & Abstract Analysis

• note

  Complex analysis

  ∑ Mathematics · College Senior — Real & Abstract Analysis

• topic

  Graduate — Master's & PhD Mathematics

  ∑ Mathematics · Graduate — Master's & PhD Mathematics

• note

  Functional analysis, PDEs

  ∑ Mathematics · Graduate — Master's & PhD Mathematics

• note

  Advanced probability & stochastic processes

  ∑ Mathematics · Graduate — Master's & PhD Mathematics

• note

  Numerical methods & optimization

  ∑ Mathematics · Graduate — Master's & PhD Mathematics

• note

  Research seminar & dissertation

  ∑ Mathematics · Graduate — Master's & PhD Mathematics

• topic

  Discrete Mathematics

  ∑ Mathematics · Discrete Mathematics

• note

  Foundations of formal logic, truth tables, and propositional calculus

  ∑ Mathematics · Discrete Mathematics

• note

  Set theory terminology, operations, and Venn diagram representations

  ∑ Mathematics · Discrete Mathematics

• note

  Counting techniques including permutations and combinations

  ∑ Mathematics · Discrete Mathematics

• note

  Graph theory basics, including paths, cycles, and connectivity

  ∑ Mathematics · Discrete Mathematics

• note

  Methods of mathematical proof such as induction and contradiction

  ∑ Mathematics · Discrete Mathematics

• equation

  P(n, r) = n! / (n - r)!

  ∑ Mathematics · Discrete Mathematics

• equation

  C(n, r) = n! / (r!(n - r)!)

  ∑ Mathematics · Discrete Mathematics

• equation

  2^n (Number of subsets in a power set)

  ∑ Mathematics · Discrete Mathematics

• equation

  V - E + F = 2 (Euler's formula for planar graphs)

  ∑ Mathematics · Discrete Mathematics

• equation

  p -> q is equivalent to ~p v q

  ∑ Mathematics · Discrete Mathematics

• topic

  Complex Analysis

  ∑ Mathematics · Complex Analysis

• note

  Holomorphic functions and the Cauchy-Riemann equations

  ∑ Mathematics · Complex Analysis

• note

  Complex integration along paths and Cauchy's Theorem

  ∑ Mathematics · Complex Analysis

• note

  Power series expansions, Taylor and Laurent series

  ∑ Mathematics · Complex Analysis

• note

  Singularities, residues, and the Residue Theorem

  ∑ Mathematics · Complex Analysis

• note

  Conformal mappings and fractional linear transformations

  ∑ Mathematics · Complex Analysis

• note

  Applications to real integrals and harmonic functions

  ∑ Mathematics · Complex Analysis

• equation

  ux = vy, uy = -vx

  ∑ Mathematics · Complex Analysis

• equation

  f(a) = (1/2pi*i) * integral(f(z)/(z-a) dz)

  ∑ Mathematics · Complex Analysis

• equation

  exp(i*theta) = cos(theta) + i*sin(theta)

  ∑ Mathematics · Complex Analysis

• equation

  integral(f(z) dz) = 2pi*i * sum(Residues)

  ∑ Mathematics · Complex Analysis

• equation

  f'(z) = lim[h->0] (f(z+h) - f(z))/h

  ∑ Mathematics · Complex Analysis

• topic

  Pre-Algebra — Foundations

  ∑ Mathematics · Pre-Algebra — Foundations

• note

  Master arithmetic operations and the properties of real numbers

  ∑ Mathematics · Pre-Algebra — Foundations

• note

  Simplify expressions using order of operations (PEMDAS)

  ∑ Mathematics · Pre-Algebra — Foundations

• note

  Solve basic linear equations and inequalities with one variable

  ∑ Mathematics · Pre-Algebra — Foundations

• note

  Understand fractions, decimals, and percentages in context

  ∑ Mathematics · Pre-Algebra — Foundations

• note

  Introductory geometry concepts and coordinate plane basics

  ∑ Mathematics · Pre-Algebra — Foundations

• note

  Translating verbal phrases into mathematical expressions

  ∑ Mathematics · Pre-Algebra — Foundations

• equation

  a + b = b + a

  ∑ Mathematics · Pre-Algebra — Foundations

• equation

  a(b + c) = ab + ac

  ∑ Mathematics · Pre-Algebra — Foundations

• equation

  x + a = b => x = b - a

  ∑ Mathematics · Pre-Algebra — Foundations

• equation

  Area = length * width

  ∑ Mathematics · Pre-Algebra — Foundations

• equation

  a/b = c/d => ad = bc

  ∑ Mathematics · Pre-Algebra — Foundations

• topic

  Partial Differential Equations

  ∑ Mathematics · Partial Differential Equations

• note

  Classification into elliptic, parabolic, and hyperbolic types

  ∑ Mathematics · Partial Differential Equations

• note

  Method of Characteristics for first-order linear and quasilinear equations

  ∑ Mathematics · Partial Differential Equations

• note

  Separation of Variables and Fourier Series for boundary value problems

  ∑ Mathematics · Partial Differential Equations

• note

  Fundamental solutions including Green's functions and the Heat Kernel

  ∑ Mathematics · Partial Differential Equations

• note

  The Well-posedness criteria of existence, uniqueness, and stability

  ∑ Mathematics · Partial Differential Equations

• equation

  u_tt = c^2 * u_xx (Wave Equation)

  ∑ Mathematics · Partial Differential Equations

• equation

  u_t = k * u_xx (Heat Equation)

  ∑ Mathematics · Partial Differential Equations

• equation

  u_xx + u_yy = 0 (Laplace Equation)

  ∑ Mathematics · Partial Differential Equations

• equation

  u(x,t) = f(x-ct) + g(x+ct) (d'Alembert Solution)

  ∑ Mathematics · Partial Differential Equations

• equation

  A*u_xx + B*u_xy + C*u_yy + ... = 0 (Second-order classification)

  ∑ Mathematics · Partial Differential Equations

• topic

  Real Analysis

  ∑ Mathematics · Real Analysis

• note

  Rigorous foundations of the real number system and its properties

  ∑ Mathematics · Real Analysis

• note

  Formal epsilon-delta definitions of limits and continuity

  ∑ Mathematics · Real Analysis

• note

  Sequences, series, and various modes of convergence

  ∑ Mathematics · Real Analysis

• note

  Differentiation and the Riemann integral from first principles

  ∑ Mathematics · Real Analysis

• note

  Point-set topology on the real line and metric spaces

  ∑ Mathematics · Real Analysis

• equation

  Axion of Completeness: Every nonempty bounded set has a supremum

  ∑ Mathematics · Real Analysis

• equation

  Definition of Limit: |f(x) - L| < epsilon whenever 0 < |x - c| < delta

  ∑ Mathematics · Real Analysis

• equation

  Cauchy Criterion: |a_n - a_m| < epsilon for all n, m > N

  ∑ Mathematics · Real Analysis

• equation

  Fundamental Theorem: Integral from a to b of f' equals f(b) - f(a)

  ∑ Mathematics · Real Analysis

• equation

  Uniform Convergence: sup |f_n(x) - f(x)| approaches 0 as n goes to infinity

  ∑ Mathematics · Real Analysis

• topic

  Linear Algebra

  ∑ Mathematics · Linear Algebra

• note

  Systems of linear equations and Gaussian elimination techniques

  ∑ Mathematics · Linear Algebra

• note

  Vector spaces, subspaces, spanning sets, and linear independence

  ∑ Mathematics · Linear Algebra

• note

  Matrix operations, determinants, and the invertible matrix theorem

  ∑ Mathematics · Linear Algebra

• note

  Linear transformations and their representation by matrices

  ∑ Mathematics · Linear Algebra

• note

  Eigenvalues, eigenvectors, and the diagonalization of matrices

  ∑ Mathematics · Linear Algebra

• note

  Inner products, orthogonality, and the Gram-Schmidt process

  ∑ Mathematics · Linear Algebra

• equation

  Ax = b

  ∑ Mathematics · Linear Algebra

• equation

  det(A - lambda*I) = 0

  ∑ Mathematics · Linear Algebra

• equation

  dot(u, v) = u^T * v

  ∑ Mathematics · Linear Algebra

• equation

  proj_L(y) = ((y*u)/(u*u)) * u

  ∑ Mathematics · Linear Algebra

• equation

  A = PDP^-1

  ∑ Mathematics · Linear Algebra

• equation

  rank(A) + nullity(A) = n

  ∑ Mathematics · Linear Algebra

• topic

  Algebra I — Linear Equations & Functions

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• note

  Identify and interpret slopes and y-intercepts in various contexts.

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• note

  Solve and graph linear equations in multiple equivalent forms.

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• note

  Define functions using domain, range, and function notation.

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• note

  Model real-world scenarios through linear modeling and data trends.

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• note

  Analyze the relationships between parallel and perpendicular lines.

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• equation

  y = mx + b

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• equation

  y - y1 = m(x - x1)

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• equation

  Ax + By = C

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• equation

  m = (y2 - y1) / (x2 - x1)

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• equation

  f(x) = ax + b

  ∑ Mathematics · Algebra I — Linear Equations & Functions

• topic

  Trigonometry — Triangles & Unit Circle

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• note

  Defining sine, cosine, and tangent via right triangle ratios (SOH CAH TOA).

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• note

  Mapping the unit circle to extend functions beyond 90-degree angles.

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• note

  Converting between degrees and radians for circular measurement.

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• note

  Applying the Law of Sines and Law of Cosines to non-right triangles.

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• note

  Using the Pythagorean Identity to relate trigonometric functions.

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• equation

  sin(theta) = opposite / hypotenuse

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• equation

  cos(theta) = adjacent / hypotenuse

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• equation

  tan(theta) = sin(theta) / cos(theta)

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• equation

  x^2 + y^2 = 1 (Unit Circle)

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• equation

  sin^2(theta) + cos^2(theta) = 1

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• equation

  c^2 = a^2 + b^2 - 2ab cos(C)

  ∑ Mathematics · Trigonometry — Triangles & Unit Circle

• topic

  Geometry — Euclidean Plane & Solid

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• note

  Axiomatic foundations of Euclidean geometry and formal proofs

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• note

  Properties of triangles, quadrilaterals, and regular polygons

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• note

  Circle theorems including chords, tangents, and arc lengths

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• note

  Area and volume of 3D solids like prisms, cones, and spheres

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• note

  Similarity, congruence, and the Pythagorean relationship

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• note

  Transformational geometry in the 2D Cartesian plane

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• equation

  c^2 = a^2 + b^2

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• equation

  Area = (1/2) * base * height

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• equation

  Volume = (1/3) * pi * r^2 * h

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• equation

  Total Surface Area = 4 * pi * r^2

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• equation

  Sum of interior angles = (n - 2) * 180

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• equation

  Arc Length = (theta/360) * 2 * pi * r

  ∑ Mathematics · Geometry — Euclidean Plane & Solid

• topic

  Precalculus — Functions & Analytic Geometry

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• note

  Comprehensive analysis of polynomial, rational, and radical functions.

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• note

  Mastery of exponential and logarithmic properties and equations.

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• note

  Geometric interpretation of conic sections: circles, parabolas, and ellipses.

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• note

  Introduction to sequences, series, and various mathematical proof methods.

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• note

  Foundational concepts of limits and continuity for calculus preparation.

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• equation

  f(x) = a(x-h)^2 + k

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• equation

  log_b(xy) = log_b(x) + log_b(y)

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• equation

  ((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• equation

  a_n = a_1 + (n-1)d

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• equation

  f(x) = (P(x))/(Q(x))

  ∑ Mathematics · Precalculus — Functions & Analytic Geometry

• topic

  Ordinary Differential Equations

  ∑ Mathematics · Ordinary Differential Equations

• note

  First-order equations: separable, linear, and exact forms

  ∑ Mathematics · Ordinary Differential Equations

• note

  Higher-order linear equations with constant coefficients

  ∑ Mathematics · Ordinary Differential Equations

• note

  Laplace transforms for solving initial value problems

  ∑ Mathematics · Ordinary Differential Equations

• note

  Systems of first-order linear equations and phase planes

  ∑ Mathematics · Ordinary Differential Equations

• note

  Power series solutions and the method of Frobenius

  ∑ Mathematics · Ordinary Differential Equations

• note

  Numerical methods including Euler and Runge-Kutta techniques

  ∑ Mathematics · Ordinary Differential Equations

• equation

  dy/dx + P(x)y = Q(x)

  ∑ Mathematics · Ordinary Differential Equations

• equation

  ay'' + by' + cy = 0

  ∑ Mathematics · Ordinary Differential Equations

• equation

  y(x) = c1*exp(r1*x) + c2*exp(r2*x)

  ∑ Mathematics · Ordinary Differential Equations

• equation

  L{f(t)} = integral[0 to inf] e^(-st)f(t)dt

  ∑ Mathematics · Ordinary Differential Equations

• equation

  x' = Ax

  ∑ Mathematics · Ordinary Differential Equations

• topic

  Algebra II — Quadratics, Polynomials, Exponentials

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• note

  Factoring, expanding, and graphing quadratic functions across multiple forms.

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• note

  Analyzing polynomial behavior including end behavior and complex roots.

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• note

  Modeling growth and decay using exponential functions and logarithms.

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• note

  Applying the Remainder Theorem and Synthetic Division to solve equations.

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• note

  Transforming functions and identifying intercepts, vertices, and asymptotes.

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• equation

  x = [-b +/- sqrt(b^2 - 4ac)] / 2a

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• equation

  y = a(x - h)^2 + k

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• equation

  f(x) = ab^x

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• equation

  log_b(y) = x iff b^x = y

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• equation

  (a + b)^n = sum(nCk * a^(n-k) * b^k)

  ∑ Mathematics · Algebra II — Quadratics, Polynomials, Exponentials

• topic

  Abstract Algebra (Groups, Rings, Fields)

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• note

  Symmetry and structure of groups including subgroups and quotients

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• note

  Homomorphisms, isomorphisms, and the fundamental mapping theorems

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• note

  Ring theory focusing on ideals, integral domains, and polynomials

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• note

  Field extensions, splitting fields, and an introduction to Galois theory

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• note

  Classification of finite abelian groups and Sylow theorems

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• equation

  G/N = { gN | g in G }

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• equation

  a * b = b * a (Commutative Law)

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• equation

  phi(ab) = phi(a)phi(b)

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• equation

  x^2 - 1 = (x - 1)(x + 1)

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• equation

  [F:K] = [F:E][E:K]

  ∑ Mathematics · Abstract Algebra (Groups, Rings, Fields)

• topic

  Number Theory

  ∑ Mathematics · Number Theory

• note

  Divisibility, prime factorization, and the Euclidean Algorithm.

  ∑ Mathematics · Number Theory

• note

  Modular arithmetic and the properties of congruences.

  ∑ Mathematics · Number Theory

• note

  Linear Diophantine equations and the Chinese Remainder Theorem.

  ∑ Mathematics · Number Theory

• note

  Arithmetic functions including Euler's Totient and Mobius inversion.

  ∑ Mathematics · Number Theory

• note

  Quadratic residues and the Law of Quadratic Reciprocity.

  ∑ Mathematics · Number Theory

• note

  Primitive roots and applications in modern cryptography.

  ∑ Mathematics · Number Theory

• equation

  a = bq + r

  ∑ Mathematics · Number Theory

• equation

  gcd(a, b) = ax + by

  ∑ Mathematics · Number Theory

• equation

  a^(p-1) == 1 (mod p)

  ∑ Mathematics · Number Theory

• equation

  phi(n) = n * Product(1 - 1/p)

  ∑ Mathematics · Number Theory

• equation

  x^2 == a (mod p)

  ∑ Mathematics · Number Theory

• equation

  a^phi(n) == 1 (mod n)

  ∑ Mathematics · Number Theory

• topic

  Topology

  ∑ Mathematics · Topology

• note

  Point-set topology: open sets, neighborhoods, and bases

  ∑ Mathematics · Topology

• note

  Continuity and homeomorphism as structural equivalence

  ∑ Mathematics · Topology

• note

  Compactness, connectedness, and separation axioms

  ∑ Mathematics · Topology

• note

  Metric spaces and the metrization problem

  ∑ Mathematics · Topology

• note

  Product and quotient topologies for complex constructions

  ∑ Mathematics · Topology

• note

  Fundamental groups and basics of algebraic topology

  ∑ Mathematics · Topology

• equation

  f: X -> Y is continuous if inverse image of any open set is open

  ∑ Mathematics · Topology

• equation

  cl(A) = A U {limit points of A}

  ∑ Mathematics · Topology

• equation

  A space is T2 if for all x, y there exist disjoint open U, V

  ∑ Mathematics · Topology

• equation

  K is compact if every open cover has a finite subcover

  ∑ Mathematics · Topology

• equation

  The product topology is generated by sets of form product(Ui)

  ∑ Mathematics · Topology

• topic

  Probability Theory

  ∑ Mathematics · Probability Theory

• note

  Axiomatic foundations: Sample spaces, events, and Kolmogorov's Three Axioms.

  ∑ Mathematics · Probability Theory

• note

  Conditional probability, independence, and the application of Bayes' Theorem.

  ∑ Mathematics · Probability Theory

• note

  Discrete and continuous random variables with their distribution functions.

  ∑ Mathematics · Probability Theory

• note

  Expectation, variance, and higher-order moments of distributions.

  ∑ Mathematics · Probability Theory

• note

  Limit theorems including the Law of Large Numbers and Central Limit Theorem.

  ∑ Mathematics · Probability Theory

• equation

  P(A|B) = P(B|A)P(A) / P(B)

  ∑ Mathematics · Probability Theory

• equation

  E[X] = sum(x * P(x)) or integral(x * f(x) dx)

  ∑ Mathematics · Probability Theory

• equation

  Var(X) = E[X^2] - (E[X])^2

  ∑ Mathematics · Probability Theory

• equation

  P(A union B) = P(A) + P(B) - P(A intersect B)

  ∑ Mathematics · Probability Theory

• equation

  M_X(t) = E[e^(tX)]

  ∑ Mathematics · Probability Theory

• topic

  Mathematical Statistics

  ∑ Mathematics · Mathematical Statistics

• note

  Foundations of point estimation including unbiasedness and efficiency

  ∑ Mathematics · Mathematical Statistics

• note

  Hypothesis testing frameworks and Neyman-Pearson Lemma

  ∑ Mathematics · Mathematical Statistics

• note

  The construction and interpretation of confidence intervals

  ∑ Mathematics · Mathematical Statistics

• note

  Likelihood-based inference and maximum likelihood estimation

  ∑ Mathematics · Mathematical Statistics

• note

  Properties of sufficient, complete, and ancillary statistics

  ∑ Mathematics · Mathematical Statistics

• note

  Bayesian vs. Frequentist approaches to statistical inference

  ∑ Mathematics · Mathematical Statistics

• equation

  L(theta) = PRODUCT f(xi; theta)

  ∑ Mathematics · Mathematical Statistics

• equation

  MSE(T) = Var(T) + [Bias(T, theta)]^2

  ∑ Mathematics · Mathematical Statistics

• equation

  Var(T) >= 1 / [n * I(theta)]

  ∑ Mathematics · Mathematical Statistics

• equation

  Lambda = sup L(theta_0) / sup L(theta)

  ∑ Mathematics · Mathematical Statistics

• equation

  P(theta | x) proportional to f(x | theta) * pi(theta)

  ∑ Mathematics · Mathematical Statistics

• topic

  Mathematical Logic & Set Theory

  ∑ Mathematics · Mathematical Logic & Set Theory

• note

  Propositional and first-order predicate logic foundations

  ∑ Mathematics · Mathematical Logic & Set Theory

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