Mathematics: GRE Math Subject Test
Mathematics is the single content domain of the GRE Math Subject Test. The exam samples undergraduate mathematics broadly: roughly half calculus, a quarter algebra, and a quarter additional topics such as discrete mathematics, elementary real analysis, geometry, and probability. Questions reward exact definitions, short proofs, and recognition of standard counterexamples, not rote computation alone.
Core rules and concepts
- Calculus: limits may be one-sided or multivariable; a multivariable limit exists only if every path gives the same value. Know derivative tests, integration by parts, improper integrals, and series convergence, including endpoint checking for power series.
- Algebra: group theory tests subgroup criteria, Lagrange’s theorem, normality, quotient groups, and field extensions. Linear algebra tests rank-nullity, eigenvalues, diagonalizability, and inner products.
- Discrete mathematics and probability: use bijections and inclusion-exclusion for counting; distinguish labeled and unlabeled objects; apply conditional probability, independence, and linearity of expectation.
- Real analysis: epsilon-delta limits, completeness, compactness, continuity, and uniform convergence; know why converses fail.
Worked example
Evaluate ∫ from 0 to 1 of x ln x dx. The integrand is unbounded near 0, so treat it as an improper integral. Integrate by parts with u = ln x and dv = x dx, giving v = x²/2. Then ∫ x ln x dx = (x²/2) ln x − x²/4 + C. At x = 1, the boundary term is 0 and the remaining term is −1/4. As x → 0⁺, x² ln x → 0, so the lower boundary contributes 0. The value is −1/4.
Traps that cost marks
Do not assume a convergent series converges absolutely; check alternating-series conditions separately. For power series, test both endpoints after finding the radius. A function can be continuous but not differentiable, and Lagrange’s converse is false. In probability, P(A|B) is not P(A and B); in counting, swapping labels changes the answer. For multivariable limits, two matching paths do not prove existence; use inequalities or polar bounds. In linear algebra, real diagonalizability requires enough real eigenvectors, not just a nonzero determinant.