GRE Mathematics Subject Test

• Integration techniques and improper integrals

  ≈ 4.5%1 h 30 min

  Fast, reliable antidifferentiation by every standard method, symmetry shortcuts for definite integrals, and convergence of improper integrals.

  • Evaluate integrals by substitution, integration by parts (including tabular and cyclic cases), partial fractions and trigonometric substitution
  • Integrate powers and products of trigonometric functions, and simplify definite integrals using odd/even symmetry and the substitution $x \mapsto a + b - x$
  • Evaluate improper integrals over infinite intervals or with unbounded integrands, and decide convergence by comparison with $\int_1^\infty x^{-p}\,dx$ and $\int_0^1 x^{-p}\,dx$
  • Apply standard results such as $\int_0^\infty x^n e^{-x}\,dx = n!$, $\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$ and reduction formulas

• The fundamental theorem, Riemann sums and integral-defined functions

  ≈ 3%1 h

  Both parts of the fundamental theorem of calculus, functions defined by integrals, limits of sums as Riemann sums and the properties of definite integrals.

  • State and apply both parts of the fundamental theorem of calculus
  • Differentiate functions defined by integrals with variable limits, e.g. $\frac{d}{dx}\int_{a(x)}^{b(x)} f(t)\,dt$, and differentiate under the integral sign
  • Recognise limits of sums as Riemann sums and evaluate them as definite integrals
  • Use linearity, additivity, comparison bounds, the mean value theorem for integrals and average value to evaluate or estimate integrals

• Applications of integration

  ≈ 3.5%1 h 15 min

  Areas, volumes, arc lengths, surface areas and physical quantities computed with single integrals in Cartesian, parametric and polar form.

  • Compute areas between curves given in Cartesian, parametric and polar form
  • Compute volumes of solids of revolution by the disc, washer and shell methods, and volumes of solids with known cross-sections
  • Compute arc length and area of surfaces of revolution
  • Apply integrals to average value, work, mass, centroids and centres of mass

• Infinite series, power series and Taylor series

  ≈ 4.5%1 h 40 min

  Convergence tests, radius and interval of convergence, building and using Taylor series, summing series in closed form and error estimates.

  • Determine convergence with the divergence, comparison, limit comparison, ratio, root, integral and alternating series tests, and distinguish absolute from conditional convergence
  • Find the radius and interval of convergence of a power series, including behaviour at the endpoints
  • Derive Taylor and Maclaurin series from the standard series for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $\arctan x$, $1/(1-x)$ and $(1+x)^\alpha$ by substitution and term-by-term differentiation or integration
  • Sum series in closed form: geometric, telescoping, and known power series evaluated at a point (e.g. $\sum n x^n$, $\sum x^n/n!$)
  • Use Taylor polynomials to approximate values, find limits and specific derivatives at a point, and bound the error with the Lagrange remainder or the alternating series estimate

About 5 h 25 min of study, lessons and core practice

• Vectors, lines, planes and curves in space

  ≈ 2.5%1 h

  Vector algebra and three-dimensional analytic geometry: products, lines and planes, distances, quadric surfaces and space curves.

  • Compute dot and cross products and use them for angles, projections, areas of parallelograms and triangles, and volumes via the scalar triple product
  • Write equations of lines and planes; find intersections, angles and distances (point to plane, point to line, between skew lines)
  • Identify cylinders and quadric surfaces (ellipsoids, paraboloids, hyperboloids, cones) from their equations
  • Analyse vector-valued functions: velocity, acceleration, unit tangent vector, arc length and curvature

• Partial derivatives, gradients and tangent planes

  ≈ 3.5%1 h 15 min

  Differential calculus of functions of several variables: partials, the chain rule, gradients and directional derivatives, tangent planes, and limits and differentiability in two variables.

  • Compute first, higher-order and mixed partial derivatives and apply Clairaut’s theorem on equality of mixed partials
  • Apply the multivariable chain rule and implicit differentiation for functions of several variables
  • Use the gradient to find directional derivatives, the direction and rate of steepest ascent and normal vectors to level curves and surfaces
  • Find tangent planes and linear approximations to graphs $z = f(x, y)$ and level surfaces $F(x, y, z) = c$
  • Decide existence of limits, continuity and differentiability for functions of two variables, including path-dependent limits

• Optimization in several variables and Lagrange multipliers

  ≈ 3%1 h

  Critical points and the second-derivative test, absolute extrema on closed regions, and constrained optimization with Lagrange multipliers.

  • Find critical points of $f(x, y)$ and classify them as local maxima, minima or saddle points with the Hessian (second-derivative) test
  • Find absolute extrema on closed, bounded regions by checking interior critical points and every piece of the boundary
  • Apply Lagrange multipliers to optimize a function subject to one constraint, and set up problems with two constraints
  • Solve geometric optimization problems such as closest points and largest boxes, using symmetry or inequalities (AM-GM, Cauchy–Schwarz) to shortcut the calculus

• Double and triple integrals and change of variables

  ≈ 3.5%1 h 20 min

  Setting up and evaluating multiple integrals over general regions, reversing the order of integration, and polar, cylindrical, spherical and general changes of variables.

  • Evaluate double and triple integrals over general regions, choosing a convenient order of integration
  • Reverse the order of integration to evaluate iterated integrals that cannot be done in the given order
  • Change variables to polar, cylindrical and spherical coordinates, and use the Jacobian determinant for general substitutions
  • Apply multiple integrals to area, volume, mass, average value and probability over regions

• Line and surface integrals and the theorems of vector calculus

  ≈ 2.5%1 h 20 min

  Line integrals, conservative fields, Green’s theorem, surface integrals and flux, divergence and curl, and the divergence and Stokes’ theorems.

  • Evaluate line integrals of scalar functions and vector fields along parametrised curves
  • Identify conservative vector fields (zero curl on a simply connected domain), find potential functions and apply the fundamental theorem for line integrals
  • Apply Green’s theorem to line integrals around closed plane curves and to compute areas
  • Compute divergence and curl, parametrise surfaces, and evaluate surface integrals and flux
  • Apply the divergence theorem and Stokes’ theorem to convert between flux, volume and circulation integrals

About 5 h 55 min of study, lessons and core practice

• First-order differential equations and models

  ≈ 2.5%1 h 10 min

  Separable, linear, exact and Bernoulli equations, initial value problems and the standard growth, cooling, mixing and logistic models.

  • Solve separable, linear (integrating factor), exact and Bernoulli first-order equations
  • Solve initial value problems and determine where the solution exists
  • Model exponential growth and decay, Newton’s law of cooling, mixing problems and logistic growth
  • Find equilibrium solutions of autonomous equations, decide their stability and interpret slope fields

• Higher-order linear equations and linear systems

  ≈ 2%1 h 10 min

  Constant-coefficient linear ODEs, particular solutions, the structure of solution spaces and first-order linear systems solved with eigenvalues.

  • Solve homogeneous linear ODEs with constant coefficients from the characteristic equation (distinct, repeated and complex roots)
  • Find particular solutions by undetermined coefficients or variation of parameters, including resonance cases
  • Use superposition, the Wronskian and the dimension of the solution space of an nth-order linear equation
  • Solve systems $\mathbf{x}' = A\mathbf{x}$ using eigenvalues and eigenvectors and describe the long-term behaviour of solutions
  • Analyse simple harmonic and damped oscillator models

About 2 h 20 min of study, lessons and core practice

• Elementary algebra: polynomials, equations and inequalities

  ≈ 2.5%1 h

  The algebraic manipulation used throughout the test: polynomials and their roots, equations and inequalities, progressions, the binomial theorem and complex arithmetic.

  • Manipulate polynomial, rational, exponential and logarithmic expressions and solve equations and inequalities, including those with absolute values
  • Apply the factor and remainder theorems, the rational root theorem and Vieta’s formulas relating roots to coefficients
  • Use the binomial theorem and sum arithmetic and geometric progressions
  • Solve nonlinear systems and simple functional equations
  • Carry out complex-number arithmetic and use the fact that non-real roots of real polynomials come in conjugate pairs

• Matrices, determinants and linear systems

  ≈ 3.5%1 h 15 min

  Matrix algebra, row reduction and the solution structure of linear systems, determinants and the many equivalent conditions for invertibility.

  • Perform matrix algebra: products, transposes, inverses, powers and block matrices
  • Solve linear systems by row reduction and relate rank, pivots and free variables to the number of solutions
  • Compute determinants by cofactor expansion and row operations, and apply properties such as $\det(AB) = \det A \det B$ and $\det(cA) = c^n \det A$
  • Apply the invertible matrix theorem and Cramer’s rule, and relate rank to the dimensions of the row, column and null spaces

• Vector spaces and linear transformations

  ≈ 4%1 h 30 min

  Subspaces, span, independence, bases and dimension; kernels, images and rank–nullity; matrices of linear maps, change of basis, and inner-product geometry.

  • Decide whether subsets are subspaces and test for span, linear independence and bases, including in spaces of polynomials, matrices and functions
  • Find the kernel and image of a linear transformation and apply the rank–nullity theorem
  • Represent linear transformations by matrices relative to given bases, change basis and recognise similar matrices
  • Use inner products, orthogonality, orthogonal projections, orthogonal complements and the Gram–Schmidt process
  • Compute dimensions of sums and intersections of subspaces using $\dim(U + W) = \dim U + \dim W - \dim(U \cap W)$

• Eigenvalues, eigenvectors and diagonalization

  ≈ 3.5%1 h 20 min

  Characteristic polynomials, eigenvalues and eigenvectors, diagonalizability, Cayley–Hamilton and the properties of special classes of matrices.

  • Compute characteristic polynomials, eigenvalues and eigenvectors, and use trace = sum and determinant = product of eigenvalues
  • Diagonalize matrices and decide diagonalizability by comparing algebraic and geometric multiplicities
  • Apply the Cayley–Hamilton theorem and minimal polynomials to compute powers and inverses of matrices
  • Use the properties of symmetric, orthogonal, rotation, projection, nilpotent and idempotent matrices
  • Compute matrix powers through eigen-decomposition and apply them to recurrences and Markov-type problems

About 5 h 5 min of study, lessons and core practice

• Groups, subgroups, cyclic and permutation groups

  ≈ 3%1 h 15 min

  Group axioms and the standard examples, element orders and Lagrange’s theorem, cyclic groups and their subgroups, and computation in symmetric groups.

  • Verify group axioms and work with standard groups: $\mathbb{Z}_n$, the unit groups $U(n)$, symmetric groups $S_n$, dihedral groups, matrix groups and the quaternion group
  • Find orders of elements and apply Lagrange’s theorem, cosets and index
  • Analyse cyclic groups: generators, the unique subgroup of each order dividing $n$ and the Euler phi function
  • Compute in permutation groups using cycle notation: products, orders from cycle structure, parity and the alternating group $A_n$

• Homomorphisms, quotients and the structure of groups

  ≈ 3%1 h 20 min

  Homomorphisms and isomorphism theorems, normal subgroups and quotients, classification of finite abelian groups and elementary Sylow theory.

  • Identify homomorphisms, kernels and images and apply the first isomorphism theorem
  • Recognise normal subgroups, form quotient groups and find centres of groups
  • Classify finite abelian groups with the fundamental theorem and count the abelian groups of a given order
  • Apply Cauchy’s theorem and elementary Sylow theorems to decide existence of subgroups and simplicity
  • Decide whether two groups are isomorphic using invariants (order, element orders, abelian or cyclic structure, direct products)

• Rings, ideals, fields and polynomials

  ≈ 2.5%1 h 10 min

  Rings, ideals, integral domains and fields; units and zero divisors; irreducibility of polynomials; finite fields and field extensions; basic modules.

  • Identify rings, subrings, ideals, integral domains and fields, including $\mathbb{Z}_n$, polynomial rings, Gaussian integers and matrix rings
  • Find units, zero divisors and the characteristic of a ring, and decide when $\mathbb{Z}_n$ is a field
  • Test polynomials for irreducibility over $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{Z}_p$ (roots for degree 2 and 3, Eisenstein’s criterion, reduction mod $p$) and describe quotient rings $F[x]/(p(x))$
  • Use basic facts about finite fields (order $p^n$, cyclic multiplicative group) and degrees of field extensions
  • Recognise elementary module concepts as generalisations of vector spaces

• Number theory

  ≈ 3%1 h 10 min

  Divisibility, gcds and the Euclidean algorithm, primes and divisor functions, congruences, and the classical theorems for computing with large powers.

  • Apply divisibility, the division algorithm, gcd and lcm, and the Euclidean algorithm with Bézout’s identity
  • Use prime factorisation to count and sum divisors and to compute Euler’s phi function
  • Solve linear congruences and linear Diophantine equations and apply the Chinese remainder theorem
  • Apply Fermat’s little theorem, Euler’s theorem and Wilson’s theorem to find remainders and last digits of large powers
  • Determine orders of units modulo $n$ and solve problems about primes, parity and divisibility of expressions

About 4 h 55 min of study, lessons and core practice

• Sequences and series of numbers and functions

  ≈ 2.5%1 h 15 min

  Completeness, monotone and Cauchy sequences, limsup and liminf, conditional convergence and rearrangements, and pointwise versus uniform convergence of functions.

  • Use the completeness axiom, supremum and infimum, and apply the monotone convergence and Bolzano–Weierstrass theorems
  • Determine convergence of sequences, including recursively defined sequences, using subsequences, Cauchy sequences, limsup and liminf
  • Distinguish absolute from conditional convergence and state the effect of rearranging a conditionally convergent series
  • Distinguish pointwise from uniform convergence of sequences and series of functions, apply the Weierstrass M-test and state what uniform convergence preserves (continuity, integrals)

• Continuity, differentiability and integrability

  ≈ 3%1 h 20 min

  The epsilon–delta definitions, uniform continuity, the big theorems on continuous and differentiable functions, Riemann integrability and the standard counterexamples.

  • Apply the ε–δ definitions of limit and continuity and the definition of uniform continuity (continuous on a compact set implies uniformly continuous)
  • Use the intermediate value, extreme value and mean value theorems to prove existence results, fixed points and inequalities
  • Relate continuity, differentiability, continuous differentiability and Lipschitz conditions, using counterexamples such as $|x|$, $x^2\sin(1/x)$ and the Dirichlet and Thomae functions
  • Decide Riemann integrability of bounded functions (continuous, monotone, finitely many discontinuities) and when limits may be interchanged with integrals and derivatives
  • Evaluate “which of the following must be true” statements about functions satisfying given hypotheses

• Topology of the real line, Euclidean space and metric spaces

  ≈ 2.5%1 h 10 min

  Open, closed, compact and connected sets in the real line and Euclidean space, limit points and closures, countability and the basic theory of metric spaces.

  • Identify open, closed, bounded, compact and connected subsets of $\mathbb{R}$ and $\mathbb{R}^n$ and apply the Heine–Borel theorem
  • Find interiors, closures, boundaries and limit points, including for sets such as $\mathbb{Q}$, $\{1/n\}$ and the Cantor set
  • Use the facts that continuous images of compact and connected sets are compact and connected, and that preimages of open sets under continuous maps are open
  • Determine countability and cardinality of sets (countable unions, $\mathbb{Q}$ countable, $\mathbb{R}$ uncountable, power sets)
  • Apply metric-space definitions: open balls, completeness, sequential compactness and equivalence of metrics

• General (point-set) topology

  ≈ 1.5%1 h

  Topologies on a set, bases, continuity and homeomorphism, and compactness, connectedness and the Hausdorff property in standard and unusual topologies.

  • Check whether a collection of subsets is a topology, and compare topologies such as discrete, indiscrete, cofinite and the usual topology
  • Determine whether functions between topological spaces are continuous and identify homeomorphisms and topological invariants
  • Decide compactness, connectedness, path-connectedness and the Hausdorff property, including in non-standard topologies
  • Work with bases, subspace topologies and product topologies

About 4 h 45 min of study, lessons and core practice

• Logic, sets, functions and relations

  ≈ 2%55 min

  Negating quantified statements, logical equivalences, set algebra, properties of functions and equivalence relations.

  • Form negations, converses and contrapositives of statements, including statements with nested quantifiers
  • Use truth tables and logical equivalences to decide validity and equivalence
  • Apply set operations, De Morgan’s laws, power sets and Cartesian products
  • Classify functions as injective, surjective or bijective, compose and invert them, and count functions between finite sets
  • Identify equivalence relations and partial orders and describe their equivalence classes

• Combinatorics and recurrence relations

  ≈ 2.5%1 h 5 min

  Counting with permutations, combinations and multisets, inclusion–exclusion and the pigeonhole principle, binomial identities and recurrences.

  • Count arrangements and selections with the multiplication principle, permutations, combinations and stars and bars
  • Apply inclusion–exclusion, derangements and the pigeonhole principle
  • Use binomial and multinomial coefficients and identities (Pascal’s rule, Vandermonde’s identity, sums of binomial coefficients)
  • Set up and solve linear recurrence relations for tiling, path-counting and Fibonacci-type problems

• Graph theory and algorithms

  ≈ 2%1 h

  Basic graph theory (degrees, trees, planarity, Euler and Hamilton circuits) and tracing algorithms given in pseudocode.

  • Use graph terminology (degree, path, cycle, connected, tree, bipartite, complete) and apply the handshake lemma
  • Apply properties of trees (a tree on n vertices has n − 1 edges) and Euler’s formula V − E + F = 2 for connected planar graphs
  • Decide whether graphs have Euler circuits or paths and recognise simple Hamiltonian and colouring arguments
  • Trace algorithms and pseudocode (loops, recursion, the Euclidean algorithm, searching and sorting) to find outputs, loop counts and orders of growth

About 3 h of study, lessons and core practice

• Complex variables

  ≈ 3%1 h 30 min

  Complex arithmetic and roots of unity, the complex exponential and logarithm, analytic functions and the Cauchy–Riemann equations, contour integration and residues.

  • Compute with complex numbers in Cartesian, polar and exponential form, apply de Moivre’s theorem and find nth roots of unity
  • Evaluate complex exponentials, logarithms (principal branch) and complex powers
  • Test analyticity with the Cauchy–Riemann equations and find harmonic conjugates
  • Evaluate contour integrals with Cauchy’s theorem, Cauchy’s integral formula and the residue theorem, including real integrals by residues
  • Apply Liouville’s theorem, the maximum modulus principle, Laurent series and the classification of isolated singularities

• Probability and statistics

  ≈ 3%1 h 15 min

  Discrete and continuous probability, conditional probability and Bayes’ theorem, standard distributions, expectation and variance, and geometric probability.

  • Compute probabilities with counting, complements, conditional probability, independence and Bayes’ theorem
  • Work with discrete (binomial, geometric, Poisson, uniform) and continuous (uniform, exponential, normal) distributions, their densities and cumulative distribution functions
  • Compute expectation, variance and standard deviation using linearity of expectation and indicator variables
  • Handle joint distributions of two variables, including geometric probability over regions of the plane
  • Interpret descriptive statistics (mean, median, standard deviation) and properties of the normal distribution

• Euclidean and analytic geometry

  ≈ 1%45 min

  Plane and solid geometry facts and coordinate methods for lengths, areas and volumes.

  • Apply plane geometry: similar triangles, the Pythagorean theorem, circle theorems, and areas of inscribed and circumscribed figures
  • Compute surface areas and volumes of prisms, cylinders, cones and spheres
  • Use coordinate methods for distances, areas (determinant or shoelace formula) and loci
  • Represent rotations and reflections of the plane as matrices and recognise isometries

• Numerical analysis

  ≈ 1%45 min

  Root-finding iterations, numerical integration rules and their errors, Euler’s method and interpolation.

  • Apply Newton’s method, bisection and fixed-point iteration, and state conditions for convergence and its order
  • Apply the trapezoidal, midpoint and Simpson’s rules, including which polynomials they integrate exactly and the sign of the error from concavity
  • Approximate solutions of initial value problems with Euler’s method
  • Use polynomial interpolation (existence and uniqueness of the interpolating polynomial) and absolute and relative error

About 4 h 15 min of study, lessons and core practice

• Pacing, elimination and no-calculator speed

  ≈ 1%45 min

  The test-taking skills behind 900+: a two-pass pacing plan, eliminating choices with special cases and counterexamples, handling Roman-numeral items and accurate mental arithmetic.

  • Pace 66 questions in 170 minutes (about 2.5 minutes each) with a two-pass strategy: bank quick calculus items, then return to flagged time sinks
  • Eliminate answer choices using special cases, counterexamples, limiting behaviour, symmetry and dimension checks
  • Handle Roman-numeral “which must be true” items by settling one statement at a time and using the structure of the answer choices
  • Carry out exact and approximate arithmetic without a calculator (e.g. $e \approx 2.718$, $\ln 2 \approx 0.693$, $\sqrt{2} \approx 1.414$)
  • Answer every question, because wrong answers are not penalised

About 45 min of study, lessons and core practice