Mathematics posters

Limits & Continuity

The limit laws, and the three conditions continuity actually requires.

Derivative Rules

Every differentiation rule a first course expects on recall.

The Chain Rule

Differentiating a composition, and the factor everyone drops.

Implicit Differentiation

Finding dy/dx for a curve where y is never isolated.

The Mean Value Theorem

The theorem behind almost every result about derivatives, hypotheses included.

L'Hopital's Rule

Resolving an indeterminate limit by differentiating top and bottom separately.

Standard Integrals

The antiderivatives worth knowing without looking anything up.

Integration by Parts

The product rule run backwards, with the rule for choosing u.

Trig Substitution

Which substitution clears which square root, and what it leaves behind.

Partial Fractions

Splitting a rational function into pieces you already know how to integrate.

Fundamental Theorem

The two halves that make differentiation and integration inverse operations.

Taylor Series

Matching a function with the polynomial whose derivatives agree at a point.

Convergence Tests

Which test to reach for, in the order worth trying them.

Partial Derivatives

Gradients, directional derivatives and what they tell you geometrically.

Vector Calculus Theorems

Green, Stokes and the divergence theorem, and what each one relates.

Matrix Multiplication

How the product is formed, which shapes are legal, and why order matters.

Gaussian Elimination

Row reduction, and how to read the answer off the echelon form.

Determinant Properties

How row operations and products change a determinant.

The Matrix Inverse

When an inverse exists, how to find one, and the order rule for products.

Vector Space Axioms

The full axiom list, in the two groups it naturally falls into.

The Subspace Test

Three checks that replace verifying all ten vector space axioms.

Basis & Dimension

How independence, spanning and dimension constrain each other.

Rank-Nullity Theorem

The conservation law relating the image and the kernel of a linear map.

Eigenvalues & Eigenvectors

The full procedure, plus the checks that catch an arithmetic slip.

Diagonalisation

Writing A as PDP inverse, and when that is not possible.

Dot & Cross Product

Two products: different outputs, different geometry, different laws.

Gram-Schmidt

Turning any basis into an orthonormal one spanning the same space.

Special Matrices

Named matrix types, their definitions and the property each one buys you.

Linear Transformations

What makes a map linear, and the four subspaces every matrix carries.

Singular Value Decomposition

The factorisation that exists for every matrix, square or not.

Truth Tables

The four core connectives on all four assignments of p and q.

Logical Equivalences

The rewrites that are valid, beside the ones that look valid and are not.

Quantifiers & Negation

How to negate a quantified statement without changing its meaning.

Proof by Induction

The five parts of a complete induction, and where marks are lost.

Proof Techniques

Six standard shapes of proof, with what each one assumes and produces.

Set Operations

The operations, and the identities worth recognising on sight.

Counting Principles

When to multiply and when to add, and the two questions that decide.

Permutations vs Combinations

The single question that separates the two, with the formulas either side.

The Binomial Theorem

The expansion of a binomial power, and where Pascal's triangle comes from.

Pigeonhole Principle

The cheapest existence argument in maths, plus its generalised form.

Inclusion-Exclusion

Counting a union without double counting the overlaps.

Recurrence Relations

Solving a linear recurrence with constant coefficients, in closed form.

Graph Vocabulary

The definitions and counting facts every graph question assumes.

Euler & Hamiltonian Paths

Two similar-sounding conditions with wildly different difficulty.

Probability Axioms

Kolmogorov's three axioms, and the results that follow from them.

Conditional Probability

Probability rescaled to the world in which one event has happened.

Bayes' Theorem

Turning a prior belief into a posterior one using the evidence.

Discrete vs Continuous

How the two kinds of random variable differ in every operation.

Expectation & Variance

The algebra of means and variances, and where independence is needed.

Discrete Distributions

The standard count distributions with their means and variances.

Continuous Distributions

The standard densities, with the first two moments of each.

Central Limit Theorem

What becomes normal, under what conditions, and what does not.

Confidence Intervals

Constructing an interval estimate, and stating what it really claims.

Significance Testing

The frequentist test in order, with what a p-value does and does not say.

Type I & Type II Errors

The two ways a test can be wrong, and the trade-off between them.

Maximum Likelihood

The estimation method behind most of statistics, in five steps.

Estimator Properties

The four properties an estimator is judged on, and how they interact.

Covariance & Correlation

Two measures of joint variation, and the dependence neither one sees.

Epsilon-Delta Limits

The definition of a limit, with the order of the quantifiers made explicit.

Sequence Convergence

What convergence requires, and the intuitions that do not survive it.

Supremum & Infimum

Least upper bounds, and the property that separates the reals from the rationals.

Cauchy Sequences

Convergence without naming a limit, and what completeness buys you.

Bolzano-Weierstrass

Every bounded sequence has a convergent subsequence, and what that does not say.

Continuity Theorems

The four results that a closed bounded interval buys you.

Uniform Continuity

One delta for the whole domain, and why compactness gives it free.

Differentiable vs Continuous

One implies the other, in exactly one direction.

Riemann Integrability

When upper and lower sums meet, and when they never do.

Absolute vs Conditional

Two kinds of convergent series, one of which survives rearrangement.

Radius of Convergence

Finding where a power series converges, endpoints included.

Uniform Convergence

What uniform convergence preserves and pointwise convergence loses.

Open & Closed Sets

The two definitions, their closure properties, and why they are not opposites.

Compactness

The equivalent characterisations, and where the familiar one stops working.

Group Axioms

The four axioms, and the facts that follow before you prove anything else.

The Subgroup Test

One condition that replaces re-checking every group axiom.

Cyclic Groups

The simplest groups there are, and the complete list of their subgroups.

Lagrange's Theorem

The divisibility constraint that rules out most candidate subgroups.

Cosets & Normal Subgroups

What cosets always do, and the extra condition that makes them a group.

Quotient Groups

Constructing G/H, and why normality is what makes it well defined.

Group Homomorphisms

Structure-preserving maps, and the kernel test for injectivity.

First Isomorphism Theorem

The result that identifies every homomorphic image as a quotient.

The Symmetric Group

Permutations in cycle notation, with order and parity read straight off.

Common Finite Groups

The standard examples every conjecture should be tested against first.

Rings, Domains & Fields

The ladder of ring structures, one added requirement per rung.

Ideals & Quotient Rings

What an ideal absorbs, and which ones give domains, which fields.

Polynomial Rings

What F[x] inherits from F, and how irreducibility builds new fields.

Modular Arithmetic

Arithmetic on remainders, and exactly when division is allowed.

Separable Equations

Separating the variables, and the solutions that step loses.

The Integrating Factor

The standard method for a first-order linear equation, in order.

Exact Equations

The cross-partial test, and what to do when it fails.

Second-Order Linear ODEs

The three root cases of the auxiliary equation, and their solutions.

Undetermined Coefficients

The trial form to guess for each kind of forcing term.

Variation of Parameters

The general method for a particular integral, and the Wronskian it needs.

Laplace Transforms

The transforms worth knowing, including the two derivative rules.

Solving ODEs by Laplace

The full route from an initial-value problem to y(t).

Systems of ODEs

Solving a constant-coefficient system through the eigenvalues of its matrix.

Phase Plane Classification

Reading the behaviour near an equilibrium off the eigenvalues.

Existence & Uniqueness

What guarantees a solution, what guarantees only one, and how local both are.

Classifying PDEs

Elliptic, parabolic or hyperbolic, and what each type implies.

Separation of Variables

Reducing a PDE to two ODEs, then rebuilding the solution from its modes.

Fourier Series

The coefficient formulas, the parity shortcuts, the convergence limits.