GRE · GRE Subject Test
GRE Mathematics
Papers, samples and curriculum documents for this course. · Dossiers, échantillons et documents de programme pour ce cours.
Course units and learning goals · Unités de cours et objectifs d'apprentissage
These lessons teach selected course objectives. Check the remaining coverage gaps; the material is not a complete preparation programme. · Ces leçons abordent des objectifs de cours sélectionnés. Vérifiez les lacunes restantes en couverture ; ce matériel ne constitue pas un programme d'entraînement complet.
C.1 · Single-variable calculus and applications
- Use limits, continuity and differentiability
- Apply derivatives, integrals and the fundamental theorem
- Recognise Riemann sums and distinguish them from infinite series
- continuity
- Agreement of a function value with its limit
- convergence
- Approach to a finite limiting value
C.2 · Multivariable calculus and vector analysis
- Compute partial derivatives, gradients and directional derivatives
- Use multiple integrals and coordinate changes
- Use derivative conditions to distinguish planes from curved surfaces
- gradient
- Vector of partial derivatives
- Jacobian
- The local area or volume scaling in a coordinate change
A.1 · Linear algebra
- Relate rank, nullity and solutions of linear systems
- Analyse vector spaces and linear transformations
- Compute eigenvalues, determinants and diagonalisation conditions
- rank
- Dimension of the image of a linear map
- eigenvalue
- A scalar satisfying Av=λv for a nonzero v
A.2 · Abstract algebra and number theory
- Use groups, subgroups, homomorphisms and quotient structures
- Distinguish rings, integral domains and fields
- Apply divisibility, congruences and elementary number theory
- homomorphism
- A map preserving the relevant operation
- field · champ
- A commutative ring where each nonzero element has an inverse
T.1 · Real analysis and topology
- Apply sequence and function limit definitions
- Distinguish compactness, connectedness and completeness
- Use metric-space and elementary topological reasoning
- compact
- Every open cover has a finite subcover
- supremum
- The least upper bound of a set
T.2 · Discrete mathematics, probability and numerical methods
- Use counting, recurrences, graph reasoning and logic
- Calculate probabilities and distribution properties
- Apply numerical approximation and assess error
- recurrence
- A rule linking a sequence term to earlier terms
- mutually exclusive
- Events that cannot happen together
T.3 · Complex analysis and residues
- Test complex differentiability
- Use contour integrals and residues
- Evaluate contour integrals with simple and higher-order pole residues
- analytic
- Complex differentiable throughout a neighbourhood
- residue
- Coefficient of (z−a)^−1 in a Laurent series
C.3 · Differential equations and initial conditions
- Solve separable first-order equations
- Solve constant-coefficient second-order equations
- Use initial conditions and uniqueness conditions
- initial condition
- A value of the solution or derivative specified at a starting point
- characteristic root
- A root of the polynomial governing exponential solutions
A.3 · Groups, cosets and quotient maps
- Verify group structure and compute element orders and subgroup indices
- Use kernels and images to identify quotient groups
- Distinguish normal subgroups from arbitrary subgroups
- Classify permutation conjugacy by cycle type
- coset
- A translate of a subgroup that forms one part of the coset partition
- normal subgroup
- A subgroup invariant under conjugation by every group element
A.4 · Rings, ideals and modules
- Distinguish units, zero divisors and integral domains
- Identify ideals and interpret quotient rings
- Compare modules over rings with vector spaces over fields
- Derive Boolean-ring properties without assuming commutativity
- ideal
- An additive subgroup of a ring absorbing multiplication by all ring elements
- torsion
- An element is annihilated by a nonzero scalar in the stated module
A.5 · Polynomials and field extensions
- Test polynomial irreducibility over the specified field
- Construct small finite fields from irreducible polynomials
- Use extension degrees and the tower law
- Use cyclotomic roots and coefficient relations to compute sums and products
- irreducible polynomial
- A positive-degree polynomial with no factorisation into smaller positive degrees over the stated field
- extension degree
- The dimension of an extension field as a vector space over its base field
A.6 · Congruences, divisibility and arithmetic functions
- Solve linear congruences using gcd conditions
- Combine coprime congruences with the Chinese remainder theorem
- Apply Euler's theorem only to invertible residues
- Use prime-exponent divisibility to find least admissible integers
- congruence
- Equality of residues because the modulus divides their difference
- totient
- The count of residues coprime to the positive integer modulus
T.4 · Sequences, series and uniform convergence
- Use Cauchy, monotone convergence and subsequence criteria
- Distinguish pointwise from uniform convergence of functions
- Check hypotheses before interchanging limits and integrals
- Cauchy sequence
- A sequence whose sufficiently late terms are arbitrarily close to one another
- uniform convergence
- Convergence with one error threshold index valid at every point of the domain
T.5 · Open sets, compactness and connectedness
- Compute closure, interior and boundary in a stated space
- Apply compactness and connectedness to continuous maps
- Distinguish relative topology from the ambient Euclidean topology
- relative topology
- Open subsets inherited by intersecting a subspace with ambient open sets
- connectedness
- Absence of a separation into disjoint nonempty relatively open subsets
C.4 · Taylor expansions and power-series endpoints
- Construct Taylor polynomials and control approximation error
- Determine radii of convergence and test endpoints separately
- Differentiate and integrate power series within their interval of convergence
- radius of convergence
- The distance from the series centre inside which a power series converges absolutely
- remainder
- The difference between a function and its finite approximation
C.5 · Improper integrals and geometric applications
- Define improper integrals by limits at each singular boundary
- Select area, volume and arc-length formulas from the geometry
- Separate convergence of integrals from signed cancellation
- improper integral
- An integral defined by limits at infinite or singular boundaries
- principal value
- A limit using prescribed symmetric cancellation that may exist when an ordinary improper integral diverges
C.6 · Multivariable extrema and constrained optimisation
- Classify two-variable critical points with the Hessian
- Use Lagrange multipliers with a regular constraint
- Compare extrema and attainable values under regular constraints
- Hessian
- The matrix of second partial derivatives used to study local curvature
- Lagrange multiplier
- A scalar relating objective and regular constraint gradients at a constrained extremum
C.7 · Coordinate changes and vector integral theorems
- Transform double and triple integrals with their Jacobians
- Apply Green, Stokes and divergence theorems with correct orientation
- Check domain singularities before asserting path independence
- divergence
- The sum of a vector field's coordinate-wise partial derivatives measuring local outward flow
- conservative field
- A vector field equal to a scalar potential gradient with path-independent line integrals
T.6 · Functions, inverse branches and composition
- Distinguish injectivity and surjectivity using the stated domain and codomain
- Choose and verify an inverse branch by composing in both directions
- Trace repeated function composition while preserving the domain
- bijection
- A function that is both injective and surjective between its specified sets
- involution
- A function whose composition with itself is the identity on its domain
T.7 · Set images, equivalence relations and logical negation
- Prove image identities and distinguish inclusion from equality
- Check reflexivity, symmetry and transitivity with explicit cases
- Negate implications and quantified statements without changing their scope
- preimage
- The set of inputs whose outputs lie in a specified target set
- equivalence relation
- A relation that is reflexive, symmetric and transitive
T.8 · Conditioning, Bayesian inference and sampling error
- Separate conditional, joint and independent-event probabilities
- Calculate posterior probabilities with a complete base-rate table
- Use variance and sample size to determine a sample mean standard error
- posterior probability
- A probability conditional on the observed evidence after accounting for prior proportions
- standard error
- The standard deviation of a statistic across repeated samples
T.9 · Similarity, scaling and conic distance loci
- Match triangle vertices by equal angles before forming a side ratio
- Apply length, area and volume scale factors to geometric changes
- Identify conic distance conditions and retain a signed hyperbola branch
- similarity
- Equality of corresponding angles and a common ratio of corresponding lengths
- focus
- A fixed point used in a conic distance definition
C.8 · Trigonometric phase and parametric curves
- Distinguish amplitude, angular frequency, phase angle and horizontal shift
- Eliminate a parameter while retaining its domain and tracing direction
- Use parametric derivatives without assuming a vertical tangent is stationary
- phase angle
- The angle offset inside a periodic function argument, defined modulo a full period
- parametric curve
- A curve whose coordinates are specified as functions of a shared parameter
T.10 · Vector geometry, projections and oriented area
- Use dot products to classify angles and compute projections
- Calculate triangle area and orientation using a cross product
- Construct or rule out planar dot-product sign configurations
- orthogonal
- Having a zero dot product in a real inner-product space
- cross product
- An oriented perpendicular vector in three dimensions whose magnitude is spanned parallelogram area
C.9 · Integrating factors and nonhomogeneous differential equations
- Solve first-order linear equations using an integrating factor
- Construct the homogeneous and particular parts of a constant-coefficient solution
- Handle resonance and verify the result in the original equation
- integrating factor
- A multiplier that turns a first-order linear equation into a product derivative
- particular solution
- One solution supplying the specified nonhomogeneous forcing
T.11 · Weak compositions, loop invariants and flowchart tracing
- Count indistinguishable allocations with nonnegative or positive constraints
- Trace a flowchart in statement order with correct reset behaviour
- Use a loop invariant and a progress measure to justify an algorithm
- weak composition
- An ordered allocation of a total into nonnegative integer parts
- loop invariant
- A property preserved before and after each iteration of a loop
C.10 · Implicit differentiation and the inverse Jacobian
- Differentiate a coupled implicit system as a linear system
- Use a nonzero Jacobian determinant to justify a local inverse
- Distinguish inverse partial derivatives from scalar reciprocal rules
- Jacobian matrix
- The matrix of first partial derivatives of a vector-valued map
- local inverse
- An inverse defined on neighbourhoods of a particular input and output point
A.7 · LU factorisation and nullity of composed maps
- Solve a factored linear system by forward and backward substitution
- Apply row permutations consistently when pivoting is required
- Bound the kernel dimension of a composition using image-kernel intersections
- forward substitution
- Solving a lower triangular system from its first equation downward
- nullity
- The dimension of the kernel of a linear map
C.11 · Integration by parts, order reversal and symmetry
- Derive integration-by-parts reductions with valid endpoint limits
- Reverse a double integral by reconstructing its region
- Use reflection symmetry while checking removable endpoint behaviour
- integration by parts
- An integration identity derived from the product rule with a boundary term
- reflection symmetry
- A relation between integrand values at points mirrored across an interval midpoint
C.12 · Related rates and removable quotient limits
- Translate a geometric rate into a derivative of the relevant quantity
- Derive and differentiate a spherical-cap volume formula
- Separate continuous quotient extension from differentiable extension
- related rate
- A rate obtained by differentiating the relationship between changing quantities
- removable limit
- A finite nearby limit used to fill in a missing function value continuously
T.12 · Metric completeness and closure from a basis
- Check a pullback metric using the properties of its defining map
- Decide completeness through the image of an isometry
- Determine closure using every basic neighbourhood instead of Euclidean intuition
- complete metric space
- A metric space in which every Cauchy sequence converges to a point of that space
- neighbourhood basis
- Basic neighbourhoods sufficient to test local topological properties
Preparing for this qualification · Préparation à cette qualification
- Approximately 66 multiple-choice items in 170 minutes; no separately timed sections. Calculus about 50%, algebra 25%, additional topics 25%. This is graduate-admissions testing of undergraduate mathematics, not an A-level qualification.
Teaching coverage still needed · Couverture pédagogique encore nécessaire
- Dedicated undergraduate lessons now supply exact teaching targets for all 66 reviewed GR3768 source items, including LU, integration methods, conjugacy, higher-order residues and alternative metrics/topologies. The source/key/crop review is in bank_review_full.yaml. Broader nonexhaustive official scope and complete-form original practice still require course audit; coordinator-owned native bank integration remains held. These preparation lessons do not certify undergraduate degree mastery.
Specifications and sample documents · Spécifications et documents d'échantillon
Course materials · Matériel pédagogique
Course preparation · Préparation du cours
Documents are available. Board-specific notes, assessments and interactive past-paper practice are not yet available for every course. · Les documents sont disponibles. Les notes spécifiques au conseil, les évaluations et la pratique interactive des anciens sujets ne sont pas encore disponibles pour tous les cours.
Lessons · Leçons →