Syllabus
Math syllabus
486 definitions and 4956 practice problems organized by topic. Search by name or by symbol.
Group theory Modern algebra I
Set · Subset · Cartesian product · Function · Binary operation · Group · Identity element · Inverse element · Abelian group
Lie theory Modern algebra
Continuity and homeomorphisms General topology
Metric spaces General topology
Metric space · Open ball · Open set in a metric space · Convergence in a metric space · Complete metric space
Functions of several variables Analysis
Preliminaries Analysis
Continuity (real analysis) Analysis
Continuous function · Intermediate value theorem · Extreme value theorem
Differentiation Analysis
Integration Analysis
Definite integral · Riemann sums · Fundamental theorem of calculus
Set language
Relations and functions
Ordinals and axioms Set theory
Partial order relation · Well-ordering · Transfinite induction · Axiom of choice · Zorn's lemma · Zermelo–Fraenkel axioms · Construction of the naturals · Construction of the reals
Propositional logic Logic
Proposition · Logical operator · Negation · Conjunction · Disjunction · Implication · Logical equivalence · NAND · Truth table · Tautology · Contradiction
Predicate logic Logic
Predicate · Universal quantifier · Existential quantifier · Negation of quantifiers · Order of quantifiers
Proof theory Logic
Axiomatic system · Rule of inference · Modus ponens · Formal proof · Consistency · Logical completeness
Completeness and incompleteness Logic
Riemannian geometry Differential geometry
LDDMM · Smooth diffeomorphic deformation · Minimum energy · Diffeomorphic flow · Velocity field · Time derivative · Composition and inverse
Limits Calculus
Limit of a function · One-sided limits · Limit laws · Indeterminate forms · Squeeze theorem · Special trigonometric limits · Limits at infinity · Asymptotes
Continuity Calculus
Continuous function · Intermediate value theorem · Extreme value theorem · Types of discontinuity · Composition and inverse
Derivatives Calculus
Derivative · Tangent line · Basic differentiation rules · Product rule · Quotient rule · Chain rule · Derivatives of trigonometric functions · Derivatives of exponentials and logarithms · Implicit differentiation
Applications of derivatives Calculus
Rolle's theorem · Mean value theorem · Critical points · Local maxima and minima · Concavity and inflection points · Optimization problems · Related rates · L'Hôpital's rule
Integrals Calculus
Definite integral · Riemann sums · Fundamental theorem of calculus · Antiderivative · Indefinite integral · Integration by substitution · Integration by parts · Area between curves
Systems and matrices Linear algebra
System of linear equations · Matrix · Matrix multiplication · Gaussian elimination · Inverse matrix · Transpose matrix
Determinants Linear algebra
Determinant · Cofactor and Laplace expansion · Properties of the determinant · Cramer's rule
Vector spaces Linear algebra
Vector in ℝⁿ · Vector space · Vector subspace · Linear combination and span · Linear independence · Basis and dimension · Rank of a matrix
Linear transformations Linear algebra
Linear transformation · Matrix of a linear transformation · Kernel and image · Rank–nullity theorem · Change of basis
Eigenvalues and eigenvectors Linear algebra
Eigenvalue and eigenvector · Characteristic polynomial · Eigenspace · Diagonalization
Orthogonality Linear algebra
Dot product · Vector norm · Orthogonal vectors · Orthogonal projection · Gram–Schmidt process
Probability foundations Probability and statistics
Sample space · Event · Probability measure · Classical probability · Addition rule · Permutations and combinations
Conditional probability Probability and statistics
Conditional probability · Independence of events · Law of total probability · Bayes' theorem
Random variables Probability and statistics
Random variable · Probability mass function · Probability density function · Cumulative distribution function · Expected value · Variance · Covariance
Distributions Probability and statistics
Bernoulli distribution · Binomial distribution · Poisson distribution · Uniform distribution · Exponential distribution · Normal distribution
Limit theorems Probability and statistics
Chebyshev inequality · Law of large numbers · Central limit theorem
Statistical inference Probability and statistics
Random sample · Sample mean · Estimator · Confidence interval · Hypothesis test · p-value
Basic concepts Differential equations
Differential equation · Order and linearity · Solution of a differential equation · Initial value problem · Existence and uniqueness theorem · Direction field
First-order equations Differential equations
Separable equation · First-order linear equation · Integrating factor · Exact equation · First-order homogeneous equation · Bernoulli equation
First-order models Differential equations
Equilibrium points · Exponential growth and decay · Logistic model · Newton's law of cooling
Higher-order linear equations Differential equations
Second-order linear equation · Superposition principle · Wronskian · Characteristic equation · Undetermined coefficients · Variation of parameters · Harmonic oscillator
Laplace transform Differential equations
Laplace transform · Inverse Laplace transform · Laplace transform of a derivative · Unit step function · Convolution
Systems and stability Differential equations
Eigenvalue and eigenvector · System of differential equations · Linear system and eigenvalues · Phase plane · Stability of an equilibrium
Vectors and curves in space Multivariable calculus
Cross product · Lines and planes in space · Parametric curve · Arc length · Quadric surface
Differentiation in several variables Multivariable calculus
Function of several variables · Limit of a function of several variables · Level curve · Partial derivative · Tangent plane · Differentiability in several variables · Gradient · Directional derivative · Chain rule in several variables
Optimization Multivariable calculus
Critical point of a multivariable function · Hessian matrix · Second derivative test · Lagrange multipliers · Absolute extrema on a region
Multiple integrals Multivariable calculus
Double integral · Fubini's theorem · Double integral over a general region · Triple integral · Polar coordinates · Cylindrical and spherical coordinates · Change of variables in multiple integrals
Line and surface integrals Multivariable calculus
Line integral of a scalar function · Vector field in the plane and space · Line integral of a vector field · Conservative field · Parametric surface · Surface integral
Integral theorems Multivariable calculus
Divergence · Curl · Stokes' Theorem · Green's Theorem · Divergence Theorem
Proofs and induction Discrete mathematics
Direct proof · Proof by contradiction · Mathematical induction · Strong induction
Sets and relations Discrete mathematics
Set operations · Inclusion-exclusion principle · Partial order relation · Hasse diagram
Counting Discrete mathematics
Permutations and combinations · Multiplication principle · Factorial · Permutations with repetition · Pigeonhole principle · Binomial coefficient · Binomial theorem · Combinations with repetition
Graph theory Discrete mathematics
Graph · Degree of a vertex · Path in a graph · Connected graph · Tree · Eulerian cycle · Hamiltonian cycle · Bipartite graph · Graph coloring · Planar graph
Recurrences and arithmetic Discrete mathematics
Recurrence relation · Divisibility · Greatest common divisor · Euclidean algorithm · Modular congruence · Homogeneous linear recurrence · Prime number
Divisibility and factorization Elementary number theory
Greatest common divisor · Euclidean algorithm · Prime number · Bézout's identity · Extended Euclidean algorithm · Euclid's lemma · Fundamental theorem of arithmetic · Least common multiple · Infinitude of primes · Linear Diophantine equation
Congruences Elementary number theory
Modular congruence · Complete residue system · Modular inverse · Linear congruence · Chinese remainder theorem · Fermat's little theorem · Euler's totient function · Euler's theorem · Wilson's theorem
Orders and quadratic residues Elementary number theory
Multiplicative order · Primitive root · Quadratic residue · Legendre symbol · Euler's criterion · Law of quadratic reciprocity
Arithmetic functions Elementary number theory
Multiplicative arithmetic function · Divisor functions · Perfect number · Möbius function · Möbius inversion formula
Continued fractions and Diophantine equations Elementary number theory
Continued fraction · Convergent of a continued fraction · Pell's equation · Sum of two squares
Primes and cryptography Elementary number theory
RSA cryptosystem · Fermat primality test · Prime number theorem
Groups and subgroups Modern algebra I
Subgroup · Order of a group · Order of an element · Generated subgroup · Cyclic group · Center of a group · Group homomorphism · Group isomorphism
Cosets and quotients Modern algebra I
Kernel of a homomorphism · Coset · Index of a subgroup · Lagrange's theorem · Normal subgroup · Quotient group · First isomorphism theorem · Direct product of groups
Actions and Sylow theorems Modern algebra I
Symmetric group · Cycle of a permutation · Sign of a permutation · Group action · Orbit-stabilizer theorem · Orbit and stabilizer · p-group · Sylow theorems
Special groups Modern algebra I
Classification of finite abelian groups · Dihedral group · Simple group
Rings and domains Modern algebra I
Unit of a ring · Zero divisor · Characteristic of a ring · First isomorphism theorem for rings
Polynomials and fields Modern algebra I
Greatest common divisor · Polynomial ring · Polynomial division algorithm · Euclidean domain · Irreducible element · Unique factorization domain · Field of fractions
Generating functions Enumerative combinatorics
Binomial coefficient · Multinomial coefficient · Exponential generating function
Classical combinatorial numbers Enumerative combinatorics
Inclusion-exclusion principle · Cycle of a permutation · Stirling number of the second kind · Bell number · Stirling number of the first kind · Catalan number · Derangement
Compositions and partitions Enumerative combinatorics
Integer composition · Integer partition · Partition generating function · Pentagonal number theorem
Symmetry and counting Enumerative combinatorics
Posets and lattices Enumerative combinatorics
Extremal combinatorics Enumerative combinatorics
Lebesgue measure Measure theory and integration
Lebesgue integral Measure theory and integration
Lp spaces Measure theory and integration
Minkowski inequality · Riesz–Fischer theorem · Cauchy sequence
Product and signed measures Measure theory and integration
Convexity and subgradients Continuous optimization
Optimality and least squares Continuous optimization
Orthogonal projection · Function of several variables · Gradient · Critical point of a multivariable function · Optimization problem · Local and global minima of convex functions · First-order necessary condition · Linear least squares
Gradient descent Continuous optimization
Line search · Wolfe conditions · Convergence with Lipschitz gradient
Second-order methods Continuous optimization
Linear and proximal programming Continuous optimization
Root finding Numerical methods
Intermediate value theorem · Bisection method · Fixed-point iteration · Secant method · Convergent sequence
Interpolation and splines Numerical methods
Rolle's theorem · Newton divided differences · Interpolation error · Runge phenomenon · Cubic splines
Numerical linear algebra Numerical methods
System of linear equations · Transpose matrix · Cholesky factorization · Jacobi and Gauss-Seidel methods
Numerical integration and differentiation Numerical methods
Trapezoidal rule · Simpson's rule · Gaussian quadrature · Finite differences for derivatives · Riemann integral
Numerical ODEs Numerical methods
Initial value problem · Euler's method · Local and global error · Runge-Kutta methods
Signals and LTI systems Signal processing
Linear transformation · Linear time-invariant systems · Impulse response
Fourier series and transform Signal processing
Sampling, DFT and FFT Signal processing
Z-transform and digital filters Signal processing
Discrete transfer function · Poles, zeros and stability · Zero and pole
Time-frequency and images Signal processing
Real numbers Real analysis
Ordered field · Supremum and infimum · Completeness axiom · Archimedean property · Absolute value · Density of the rationals
Sequences Real analysis
Convergent sequence · Uniqueness of limits · Monotone sequence · Monotone convergence theorem · Subsequence · Bolzano-Weierstrass theorem · Cauchy sequence
Limits and continuity Real analysis
Continuous function · Intermediate value theorem · Extreme value theorem · Limit of a function · Sequential continuity · Uniform continuity
Differentiation Real analysis
Derivative · Rolle's theorem · Mean value theorem · L'Hôpital's rule · Differentiability and continuity
Integration and series Real analysis
Fundamental theorem of calculus · Partition and Darboux sums · Riemann integral · Integrability of continuous functions · Numerical series · Comparison test · Absolute convergence · Uniform convergence
Complex numbers Complex analysis
Complex number · Complex conjugate · Modulus of a complex number · Polar form · Complex nth root · De Moivre's formula · Topology of the complex plane
Functions and limits Complex analysis
Complex function · Complex limit · Complex continuity · Complex exponential · Complex logarithm · Complex trigonometric functions
Analytic functions Complex analysis
Complex derivative · Cauchy-Riemann equations · Holomorphic function · Harmonic function · Möbius transformation
Complex integration Complex analysis
Curve and contour · Contour integral · Cauchy-Goursat theorem · Complex primitive · Cauchy integral formula · Liouville's theorem · Fundamental theorem of algebra
Series and residues Complex analysis
Complex Taylor series · Laurent series · Isolated singularity · Zero and pole · Residue · Residue theorem · Real integral by residues
Topological spaces General topology
Topology · Topological space · Discrete and trivial topology · Closed set · Basis of a topology · Neighborhood of a point · Interior of a set · Closure of a set · Boundary of a set · Accumulation point
Continuity and constructions General topology
Continuous function between topological spaces · Homeomorphism · Subspace topology · Product topology · Quotient topology
Separation and connectedness General topology
Hausdorff space · Connected space · Continuous image of a connected space · Path-connected space · Connected component
Compactness General topology
Open cover · Compact space · Compact sets in Hausdorff spaces · Continuous image of a compact space · Heine-Borel theorem · Sequential compactness · Lebesgue number
Operations and families Set theory
Subset · Set membership · Union of families · Intersection of families · Relative complement · Power set
Relations and functions Set theory
Cartesian product · Binary relation · Equivalence relation · Equivalence class · Injective function · Surjective function · Bijective function · Image and preimage · Composition and inverse
Cardinals Set theory
Finite set · Infinite set · Equipotence · Countable set · Cantor's diagonal theorem · Cardinal number · Schröder–Bernstein theorem