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Limits and Continuity
A free GRE Mathematics note on limit laws, the squeeze theorem, L'Hopital's rule, the three-condition definition of continuity, and the classic limits every calculus question assumes you know.
- Evaluate limits with the limit laws and recognize when a nonzero-denominator substitution finishes the problem immediately.
- State the three-condition definition of continuity at a point and use it to classify discontinuities.
- Apply the squeeze theorem to bounded-oscillation limits such as $x^2 \sin(1/x)$.
Formulas covered: Classic sine limit · Squeeze theorem · L'Hopital's rule
Read the Limits and Continuity noteSeries Convergence Tests
A free GRE Mathematics note on the convergence toolkit: geometric and p-series benchmarks, comparison, ratio, and root tests, the alternating series test, and the difference between absolute and conditional convergence.
- Classify geometric and p-series on sight and use them as comparison benchmarks.
- Choose between the direct comparison and limit comparison tests for positive-term series.
- Apply the ratio and root tests to series with factorials or nth powers, and state when each is inconclusive.
Formulas covered: Geometric series sum · p-series convergence · Ratio test
Read the Series Convergence Tests noteEigenvalues and Determinants
A free GRE Mathematics linear algebra note on determinant properties, the eigenvalue equation, trace and determinant as eigenvalue invariants, triangular matrices, and the basics of diagonalizability.
- Use the determinant's row properties — sign change under exchange, linearity in each row, product of diagonal entries for triangular matrices — to evaluate determinants without brute-force expansion.
- Set up and solve the eigenvalue equation $\det(A - \lambda I) = 0$ for small matrices.
- Check computed eigenvalues instantly against the trace (their sum) and the determinant (their product).
Formulas covered: 2x2 determinant · Characteristic polynomial of a 2x2 matrix · Eigenvalue invariants
Read the Eigenvalues and Determinants noteGroup Theory Essentials
A free GRE Mathematics note on group and subgroup axioms, element order, cyclic groups, Lagrange's theorem and its corollaries, direct products, and how to tell two small groups apart.
- Verify the group and subgroup axioms quickly, and name which axiom a proposed structure fails.
- Compute the order of an element in $\mathbb{Z}_n$, in a direct product, and in a permutation group.
- Use Lagrange's theorem and its corollaries to rule out impossible subgroup and element orders.
Formulas covered: Lagrange's theorem · Order of a power in a cyclic group · Order in a direct product · Elements of a given order in a cyclic group
Read the Group Theory Essentials noteMetric Spaces and Compactness
A free GRE Mathematics note on metrics, open and closed sets, the Heine-Borel characterisation of compactness, what compactness buys for continuous functions, and how connectedness is tested.
- Decide whether a given subset of $\mathbb{R}$ or $\mathbb{R}^n$ is open, closed, both, or neither, by checking interior points and limit points.
- Apply the Heine-Borel characterisation correctly and state where it stops applying.
- Produce an explicit open cover with no finite subcover to prove a set is not compact.
Formulas covered: Open ball · Heine-Borel (Euclidean space only) · Covering definition of compactness · Continuity by preimages
Read the Metric Spaces and Compactness noteFirst- and Second-Order Differential Equations
A free GRE Mathematics note on classifying an ODE, separable equations, the integrating factor for first-order linear equations, and the three characteristic-root cases for constant-coefficient second-order equations.
- Classify an equation by order and linearity, and choose separation or an integrating factor for a first-order problem.
- Solve a first-order linear equation with the integrating factor and impose an initial condition.
- Write the general solution of $ay'' + by' + cy = 0$ from the characteristic roots in all three cases.
Formulas covered: Integrating factor for a first-order linear equation · Separable equation · Characteristic equation, three cases
Read the First- and Second-Order Differential Equations noteAnalytic Functions and Residues
A free GRE Mathematics note on the Cauchy-Riemann test for analyticity, Cauchy's theorem and integral formula, classifying singularities from a Laurent series, and evaluating contour integrals with residues.
- Test a candidate function for analyticity with the Cauchy-Riemann equations and say why $\bar{z}$ fails.
- Decide when Cauchy's theorem alone forces a contour integral to vanish.
- Classify an isolated singularity as removable, a pole of stated order, or essential.
Formulas covered: Cauchy-Riemann equations · Cauchy's integral formula for derivatives · Residue at a pole of order m · Residue theorem
Read the Analytic Functions and Residues noteCalculus Domain Guide
The whole GRE Mathematics calculus domain in one place: derivatives as linear approximation, the fundamental theorem, the series toolkit, gradients and Lagrange multipliers, and the Green/Stokes/divergence family.
- Differentiate and integrate the standard library fluently, including the chain-rule factor that appears when the fundamental theorem is used with variable limits.
- Choose a convergence test from the shape of the terms — factorials and exponentials to the ratio test, $n$-th powers to the root test, rational terms to comparison with a $p$-series.
- Reconstruct $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, and $1/(1-x)$ from memory and use them to evaluate limits and integrals faster than L'Hopital's rule allows.
Formulas covered: Fundamental theorem with variable limits · Convergence benchmarks: geometric and p-series · Ratio and root tests · Maclaurin series library · Gradient, directional derivative, and tangent plane · Second-derivative test in two variables · Lagrange multipliers · Green, Stokes, and divergence theorems · Polar and spherical area/volume factors
Read the Calculus Domain Guide noteLinear Algebra Domain Guide
Vector spaces, rank and nullity, determinants, eigenvalues, diagonalisation, and linear operators for the GRE Mathematics Test — with the invertibility equivalences and the trace/determinant checks that settle most items in seconds.
- Use rank–nullity to convert a statement about solutions of $A\vec{x} = \vec{b}$ into a dimension count, and vice versa.
- Recite the invertibility equivalences and use any one of them as a substitute for a computation.
- Compute determinants by row reduction and by cofactor expansion, and apply $\det(AB) = \det A \det B$, $\det(cA) = c^n \det A$, and $\det(A^{-1}) = 1/\det A$.
Formulas covered: Rank–nullity theorem · Determinant properties · Characteristic polynomial and its invariants · Spectrum of related matrices · Diagonalisability criterion · Dimension of a sum of subspaces · Real symmetric matrices
Read the Linear Algebra Domain Guide noteAbstract Algebra Domain Guide
Groups, rings, fields, and the number theory the GRE Mathematics Test actually uses: Lagrange's theorem, cyclic structure, quotients and the first isomorphism theorem, zero divisors, and modular arithmetic.
- Use Lagrange's theorem to rule out subgroup and element orders, and state why its converse fails.
- Describe every subgroup and count the elements of each order in a cyclic group from the divisors of $n$.
- Decide whether two groups of the same order are isomorphic by comparing invariants such as abelianness and order counts.
Formulas covered: Lagrange's theorem and index · Order of a power in a cyclic group · Counts in a cyclic group of order n · Orders in a direct product · First isomorphism theorem · Standard finite groups and their orders · The ring hierarchy · Euler and Fermat congruences
Read the Abstract Algebra Domain Guide noteReal Analysis Domain Guide
Completeness, sequences, uniform versus pointwise convergence, continuity theorems, and the counting and probability rules the GRE Mathematics Test groups with them — taught through the counterexamples that decide the questions.
- State the monotone convergence theorem and the Bolzano–Weierstrass theorem and use them to prove a sequence converges without producing its limit.
- Distinguish continuity, uniform continuity, and differentiability, and give a standard function separating each pair.
- Test a sequence of functions for uniform convergence using the supremum of $|f_n - f|$.
Formulas covered: Convergence machinery for sequences · Uniform convergence test · Continuity on a compact set · Intermediate and mean value theorems · Permutations and combinations · Probability rules · Expectation and variance
Read the Real Analysis Domain Guide noteTopology Domain Guide
Point-set topology for the GRE Mathematics Test: the open-set axioms, continuity by preimages, closure and boundary, and which properties — compactness and connectedness — actually survive a continuous map.
- Apply the topology axioms, including the asymmetry between arbitrary unions and finite intersections.
- Use the preimage characterisation of continuity in place of $\varepsilon$–$\delta$ when no metric is available.
- Compute closure, interior, and boundary for standard subsets of $\mathbb{R}$ and $\mathbb{R}^2$.
Formulas covered: Topology axioms · Continuity by preimages · Heine–Borel and its limits · What continuous maps preserve · Compactness and closedness interact · Connectedness in the real line
Read the Topology Domain Guide noteDifferential Equations Domain Guide
How the GRE Mathematics differential-equations domain fits together: existence and uniqueness, the homogeneous-plus-particular structure, the Wronskian, linear systems solved by eigenvalues, and long-run behaviour from equilibria.
- Classify an equation and choose its method from the classification alone.
- State the existence-and-uniqueness hypothesis and give an equation whose solution is not unique because that hypothesis fails.
- Use the homogeneous-plus-particular structure to shortcut nonhomogeneous problems, including the resonance case where the trial solution needs an extra factor of $x$.
Formulas covered: Existence and uniqueness for a first-order IVP · Integrating factor for a first-order linear equation · Solution structure for linear equations · Undetermined coefficients with resonance · Wronskian and Abel's identity · Linear systems by eigenvalues · Stability from signs · Cauchy–Euler equation
Read the Differential Equations Domain Guide noteComplex Analysis Domain Guide
From polar form and roots of unity to Cauchy–Riemann, Cauchy's theorem, Laurent series, and residues — the complex-analysis material the GRE Mathematics Test expects, with the rigidity theorems that make its questions answerable at a glance.
- Convert between rectangular and polar form fluently and use de Moivre's theorem to find powers and $n$-th roots.
- Test analyticity with the Cauchy–Riemann equations and construct a harmonic conjugate.
- Decide whether Cauchy's theorem applies, and use Cauchy's integral formula and its derivative version when it does not.
Formulas covered: Polar form, de Moivre, and roots · Cauchy–Riemann equations · Cauchy's theorem and integral formula · Residue formulas · Residue theorem · Classification of isolated singularities · Liouville and maximum modulus
Read the Complex Analysis Domain Guide note