GRE Mathematics Reference Sheet

How to revise from this sheet

1. 1Cover the right-hand column and say the relationship out loud before reading it. Recognising a formula is not the same as recalling it, and only recall survives a timed section.
2. 2Work down a single domain at a time, then answer a few questions in that domain immediately. Recall that is never used decays fastest.
3. 3When an entry has a condition attached, rehearse the condition as part of the statement. Most exam traps are built on applying a correct formula outside its hypothesis.

ETS lists the GRE Mathematics content areas as calculus 50%, algebra 25%, and additional topics 25%, with the additional-topics questions drawn from other areas of mathematics currently studied by undergraduates.

Calculus

Half the test. Recall must be instant here, because the time saved is what pays for the algebra and analysis items.

Fundamental theorem, variable limits

$$\frac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))h'(x) - f(g(x))g'(x)$$

d/dx of the integral of f from g(x) to h(x) equals f(h(x))h'(x) - f(g(x))g'(x)

Both chain-rule factors, and the minus sign on the lower limit, are what the question is usually testing.

Integration by parts

$$\int u\,dv = uv - \int v\,du$$

integral of u dv = uv - integral of v du

Choose u by the order logarithm, inverse trig, algebraic, trigonometric, exponential — the earlier class differentiates away.

Convergence benchmarks

$$\sum_{n \ge 0} ar^{n} = \frac{a}{1-r}\ (|r|<1); \qquad \sum_{n \ge 1} n^{-p} \text{ converges} \iff p > 1$$

geometric sum a/(1-r) converges iff |r| < 1; the p-series sum 1/n^p converges iff p > 1

Everything else is compared to these two families; p = 1 is the harmonic series and diverges.

Ratio and root tests

$$\rho < 1 \Rightarrow \text{abs. convergent}, \quad \rho > 1 \Rightarrow \text{divergent}, \quad \rho = 1 \Rightarrow \text{no conclusion}$$

rho = lim |a_{n+1}/a_n| or lim |a_n|^{1/n}: below 1 absolutely convergent, above 1 divergent, exactly 1 inconclusive

Reach for these when factorials or n-th powers appear. Every p-series returns rho = 1, which is why it decides nothing.

Maclaurin series library

$$e^{x} = \sum \frac{x^{n}}{n!}, \ \sin x = \sum \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}, \ \cos x = \sum \frac{(-1)^{n}x^{2n}}{(2n)!}, \ \frac{1}{1-x} = \sum x^{n}, \ \ln(1+x) = \sum \frac{(-1)^{n+1}x^{n}}{n}$$

e^x = sum x^n/n!; sin x = sum (-1)^n x^(2n+1)/(2n+1)!; cos x = sum (-1)^n x^(2n)/(2n)!; 1/(1-x) = sum x^n; ln(1+x) = sum (-1)^(n+1) x^n/n

The single highest-yield recall object on the sheet: it answers indeterminate limits, non-elementary integrals, and coefficient questions alike.

Directional derivative and gradient

$$D_{\vec{u}}f = \nabla f \cdot \vec{u}, \quad |\nabla f| = \text{max rate}$$

D_u f = grad f . u for a unit vector u; grad f is orthogonal to the level set and points along fastest increase

Normalise the direction first. The derivative along a level curve is zero, which is a fast way to check an answer.

Second-derivative test (two variables)

$$D = f_{xx}f_{yy} - f_{xy}^{2}$$

D = f_xx f_yy - (f_xy)^2: D > 0 with f_xx > 0 minimum, D > 0 with f_xx < 0 maximum, D < 0 saddle, D = 0 inconclusive

Applies only where the gradient vanishes.

Lagrange multipliers

$$\nabla f = \lambda \nabla g, \quad g = c$$

extremise f subject to g = c by solving grad f = lambda grad g with g = c

Produces candidates only — compare f at all of them, and check the boundary when the constraint set is not closed.

Coordinate change factors

$$dA = r\,dr\,d\theta, \quad dV = r\,dr\,d\theta\,dz, \quad dV = \rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta$$

dA = r dr dtheta (polar); dV = r dr dtheta dz (cylindrical); dV = rho^2 sin(phi) drho dphi dtheta (spherical)

The Jacobian factor is part of the integral, not decoration.

Green, Stokes, divergence

$$\oint_C (P\,dx + Q\,dy) = \iint_D (Q_x - P_y)\,dA, \quad \oint_C \vec F \cdot d\vec r = \iint_S (\nabla \times \vec F)\cdot \vec n \,dS, \quad \oiint_S \vec F \cdot \vec n\,dS = \iiint_E \nabla \cdot \vec F \,dV$$

loop integral of P dx + Q dy = double integral of (Q_x - P_y) dA; loop integral of F.dr = surface integral of curl F . n dS; closed-surface flux of F = triple integral of div F dV

One pattern at three dimensions: the total of a derivative over a region equals a total over its boundary. Orientation controls the sign.

Conservative fields

$$\int_C \nabla f \cdot d\vec r = f(B) - f(A); \qquad Q_x = P_y \text{ on a simply connected domain} \iff \text{conservative}$$

F = grad f implies the line integral is f(end) - f(start) and every closed loop gives 0; on a simply connected planar domain, Q_x = P_y characterises conservative fields

Simple connectivity is a real hypothesis: the vortex field on the punctured plane satisfies Q_x = P_y and still has circulation 2 pi.

Linear algebra

Recall here is mostly equivalences and invariants; almost nothing on this list requires a computation longer than a line.

Rank–nullity

$$\operatorname{rank}(A) + \operatorname{nullity}(A) = n$$

rank(A) + nullity(A) = number of columns of A

Counted against the domain, so an m x n matrix always gives n.

Invertibility equivalences

$$A^{-1} \text{ exists} \iff \det A \neq 0 \iff \operatorname{rank} A = n \iff \ker A = \{\vec 0\} \iff 0 \notin \operatorname{spec}(A)$$

for square A: invertible <=> det A nonzero <=> rank n <=> columns independent <=> kernel trivial <=> 0 is not an eigenvalue <=> row-reduces to I

A question asked in one of these forms can be answered in any other; pick whichever is cheapest for the given data.

Determinant rules

$$\det(AB) = \det A\det B, \quad \det(cA) = c^{n}\det A, \quad \det(A^{-1}) = 1/\det A$$

det(AB) = det A det B; det(A^T) = det A; det(A^-1) = 1/det A; det(cA) = c^n det A; triangular determinant is the diagonal product

There is no rule for det(A + B). The exponent n in det(cA) is the usual slip.

Trace and determinant as spectral invariants

$$\sum \lambda_i = \operatorname{tr}A, \quad \prod \lambda_i = \det A, \quad \lambda^{2} - (\operatorname{tr}A)\lambda + \det A = 0$$

sum of eigenvalues = trace; product of eigenvalues = determinant; 2x2 characteristic polynomial is lambda^2 - (tr A) lambda + det A

Two cheap numbers that often give a 2x2 spectrum outright and always check a 3x3 answer.

Spectrum of related matrices

$$A^{k} \to \lambda^{k}, \quad A^{-1} \to \lambda^{-1}, \quad A + cI \to \lambda + c, \quad p(A) \to p(\lambda)$$

A^k has lambda^k; A^-1 has 1/lambda; A + cI has lambda + c; A^T has the same eigenvalues; p(A) has p(lambda)

Eigenvectors are unchanged by all of these except transposition. Never recompute a spectrum you already know.

Diagonalisability

$$A = S\Lambda S^{-1}, \quad A^{k} = S\Lambda^{k}S^{-1}$$

A is diagonalisable exactly when geometric multiplicity equals algebraic multiplicity for every eigenvalue; then A = S Lambda S^-1 and A^k = S Lambda^k S^-1

n distinct eigenvalues is sufficient, never necessary. The identity matrix repeats an eigenvalue and is already diagonal.

Real symmetric matrices

$$A = A^{T} \Rightarrow A = QDQ^{T}$$

real eigenvalues, eigenvectors for distinct eigenvalues orthogonal, orthogonally diagonalisable as A = Q D Q^T

Orthogonality of eigenvectors is a symmetry property; a general matrix guarantees only independence.

Dimension of a sum

$$\dim(U+W) = \dim U + \dim W - \dim(U \cap W)$$

dim(U + W) = dim U + dim W - dim(U intersect W)

The sum is direct exactly when the intersection is trivial.

Abstract algebra and number theory

Constraint facts, not computations. Most items are decided by a divisibility statement you either recall or do not.

Lagrange's theorem

$$|G| = [G:H]\,|H|, \qquad |a| \mid |G|$$

|G| = [G : H] |H|; the order of every element divides |G|; a^|G| = e; groups of prime order are cyclic

The converse is false: A_4 has order 12 and no subgroup of order 6.

Cyclic group counts

$$|a^{k}| = \frac{n}{\gcd(k,n)}, \quad \#\{|x| = d\} = \varphi(d)$$

in a cyclic group of order n: |a^k| = n/gcd(k,n); one subgroup per divisor of n; phi(d) elements of order d; phi(n) generators

The factorisation of n is the whole computation. phi(p^k) = p^k - p^(k-1), and phi is multiplicative on coprime arguments.

Direct products

$$|(g,h)| = \operatorname{lcm}(|g|,|h|), \quad \mathbb{Z}_m \times \mathbb{Z}_n \cong \mathbb{Z}_{mn} \iff \gcd(m,n)=1$$

|(g,h)| = lcm(|g|, |h|); Z_m x Z_n is cyclic exactly when gcd(m, n) = 1

Z_2 x Z_2 is not Z_4 — the standard non-isomorphic pair of order 4.

Standard groups

$$|S_n| = n!, \quad |A_n| = \tfrac{n!}{2}, \quad |D_n| = 2n$$

|S_n| = n!; |A_n| = n!/2; |D_n| = 2n; a permutation's order is the lcm of its disjoint cycle lengths; a k-cycle is even iff k is odd

Cycle structure answers order and parity questions without multiplying permutations.

First isomorphism theorem

$$G/\ker\varphi \cong \operatorname{im}\varphi$$

for a homomorphism f, G/ker(f) is isomorphic to im(f); ker(f) is normal; f is injective iff ker(f) is trivial

The quickest route to identifying a quotient group without touching cosets.

Ring hierarchy

$$\text{field} \subset \text{integral domain} \subset \text{comm. ring with }1$$

ring, then commutative ring with identity, then integral domain (no zero divisors), then field (every nonzero element a unit); every finite integral domain is a field

Cancellation is available exactly when there are no zero divisors: 2 x 3 = 0 in Z_6 blocks it.

Modular arithmetic

$$a^{\varphi(n)} \equiv 1 \pmod n, \qquad a^{p-1} \equiv 1 \pmod p$$

Z_n is a field iff n is prime; the units of Z_n are the residues coprime to n, so |U(n)| = phi(n); a^phi(n) = 1 mod n for gcd(a,n) = 1, and a^(p-1) = 1 mod p for a prime p not dividing a

These congruences are Lagrange's theorem inside the group of units — the fast route to last-digit and large-power questions.

Real analysis, counting, and probability

Hypotheses and counterexamples. Recall the condition each theorem needs, because that is the half the question removes.

Sequence convergence machinery

$$\text{bounded} + \text{monotone} \Rightarrow \text{convergent}; \quad \text{Cauchy} \iff \text{convergent in } \mathbb{R}$$

bounded and monotone implies convergent; every bounded sequence has a convergent subsequence; in R, convergent and Cauchy are the same

Cauchy implies convergent only in a complete space; in Q the approximations to the square root of 2 fail.

Continuity at a point

$$\lim_{x \to a} f(x) = f(a)$$

f is continuous at a iff f(a) is defined, the limit at a exists, and the two agree

Three separate checks; naming the one that fails classifies the discontinuity.

Continuity on a compact set

$$f \in C(K),\ K \text{ compact} \Rightarrow \max, \min \text{ attained}$$

continuous on compact implies bounded, attains a maximum and minimum, and is uniformly continuous

Drop compactness and all three fail: 1/x on the open interval (0,1) is the standing counterexample.

Uniform convergence test

$$\sup_{x \in S}|f_n(x) - f(x)| \to 0$$

f_n converges to f uniformly on S iff the supremum of |f_n - f| over S tends to 0

Uniform limits preserve continuity and commute with integration, but not with differentiation: sin(nx)/n converges uniformly to 0 while its derivative diverges.

Mean and intermediate value theorems

$$f'(c) = \frac{f(b)-f(a)}{b-a}$$

continuous on [a,b] gives every intermediate value; continuous on [a,b] and differentiable on (a,b) gives f'(c) = (f(b)-f(a))/(b-a) for some c

Both are existence statements only; neither locates the point.

Counting

$$P(n,r) = \frac{n!}{(n-r)!}, \quad \binom{n}{r} = \frac{n!}{r!(n-r)!}$$

ordered: P(n,r) = n!/(n-r)!; unordered: C(n,r) = n!/(r!(n-r)!); inclusion-exclusion: |A or B| = |A| + |B| - |A and B|

Decide in writing whether order matters before choosing. Divide out each symmetry once and only once.

Probability rules

$$P(A \cup B) = P(A) + P(B) - P(A \cap B), \quad P(A \cap B) = P(A \mid B)P(B)$$

P(A or B) = P(A) + P(B) - P(A and B); P(A and B) = P(A|B)P(B); independence means P(A and B) = P(A)P(B)

Mutually exclusive is not independent: two events of positive probability that exclude each other are dependent.

Expectation and variance

$$E[X+Y] = E[X]+E[Y], \quad \operatorname{Var}(aX+b) = a^{2}\operatorname{Var}(X)$$

E[X + Y] = E[X] + E[Y] always; Var(aX + b) = a^2 Var(X); variances add only for independent variables

Linearity of expectation never needs independence; additivity of variance always does.

Topology

One table does most of the work: what a continuous map preserves, and what it does not.

Topology axioms

$$\bigcup_\alpha U_\alpha \text{ open}, \quad U_1 \cap \cdots \cap U_n \text{ open}$$

the empty set and X are open; arbitrary unions of open sets are open; only finite intersections are

The intersection of all intervals (-1/n, 1/n) is the single point 0, which is not open.

Continuity by preimages

$$f^{-1}(U) \text{ open for all open } U$$

f is continuous iff the preimage of every open set is open, equivalently the preimage of every closed set is closed

Preimages, not images: x squared maps the open interval (-1,1) onto [0,1), which is not open.

Heine–Borel

$$K \subseteq \mathbb{R}^{n}: \ \text{compact} \iff \text{closed and bounded}$$

in Euclidean space, compact is exactly closed and bounded; in a general metric space, compact is equivalent to sequentially compact

The equivalence is special to R^n. An infinite set with the discrete metric is closed and bounded but not compact.

Preservation table

$$f(K) \text{ compact}, \quad f(C) \text{ connected}$$

continuous maps preserve compactness and connectedness; they do not preserve openness, closedness, boundedness, or completeness

The extreme and intermediate value theorems are these two rows applied to real-valued f.

Compact and closed interplay

$$C \text{ closed} \subseteq K \text{ compact} \Rightarrow C \text{ compact}$$

a closed subset of a compact space is compact; a compact subset of a Hausdorff space is closed

Every metric space is Hausdorff, so both directions are automatic there.

Connectedness in R

$$C \subseteq \mathbb{R} \text{ connected} \iff C \text{ is an interval}$$

the connected subsets of the real line are exactly the intervals; path-connected implies connected but not conversely

Combined with preservation this is the intermediate value theorem; the topologist's sine curve blocks the converse.

Differential equations

Classify, then apply the matching method. Behaviour questions are answered from signs without solving.

First-order linear

$$\mu = e^{\int p\,dx}, \qquad (\mu y)' = \mu q$$

y' + p(x)y = q(x): multiply by mu = exp(integral p), then (mu y)' = mu q

Normalise the coefficient of y' to 1 before reading off p.

Separable

$$\int \frac{dy}{h(y)} = \int g(x)\,dx$$

dy/dx = g(x)h(y): integrate dy/h(y) = g(x) dx, and add back each constant solution where h vanishes

Dividing by h(y) silently discards genuine solutions.

Characteristic roots

$$y = \begin{cases} c_1e^{r_1x} + c_2e^{r_2x} \\ (c_1 + c_2x)e^{rx} \\ e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)\end{cases}$$

ay'' + by' + cy = 0: distinct real roots give two exponentials; a repeated root r gives (c1 + c2 x)e^(rx); alpha +/- beta i gives e^(alpha x) times cosine and sine of beta x

The extra factor of x in the repeated case is the same phenomenon as resonance in the forced problem.

Nonhomogeneous structure

$$y = y_h + y_p$$

the general solution is the homogeneous general solution plus any one particular solution; if the trial form already solves the homogeneous equation, multiply it by x

Undetermined coefficients needs constant coefficients and a forcing term that is polynomial, exponential, sinusoidal, or a product of these.

Wronskian

$$W(x) = W(x_0)e^{-\int p}$$

W = y1 y2' - y1' y2 is nonzero exactly when the two solutions are independent; Abel's identity makes W either identically zero or never zero

One evaluation at a convenient point settles independence on the whole interval.

Linear systems

$$\vec x(t) = \sum_i c_i e^{\lambda_i t}\vec v_i$$

x' = Ax has solutions exp(lambda t) v for eigenpairs of A; complex eigenvalues give spirals; a deficient repeated eigenvalue introduces a t times exponential term

The eigenvalue problem for A is the entire method.

Stability from signs

$$\operatorname{Re}\lambda_i < 0 \ \forall i \Rightarrow \vec x \to \vec 0; \qquad f'(y^{*}) < 0 \Rightarrow \text{stable}$$

all eigenvalues with negative real part: asymptotically stable; any with positive real part: unstable; mixed signs: saddle; purely imaginary: centre. Scalar y' = f(y): the equilibrium y* is stable when f'(y*) < 0

A negative trace does not imply stability: negative determinant means a saddle regardless.

Existence and uniqueness

$$f, \ \tfrac{\partial f}{\partial y} \text{ continuous} \Rightarrow \text{unique}$$

y' = f(x,y) with f continuous near the initial point has a solution; continuity of df/dy as well makes it unique

y' = y^(2/3) with y(0) = 0 has both the zero solution and (x/3)^3, because the partial derivative blows up there.

Complex analysis

Rigidity theorems first, computation second: several items each cycle are answered by ruling a function out entirely.

Polar form and roots

$$z^{n} = r^{n}e^{in\theta}, \qquad w^{1/n} = \rho^{1/n}e^{i(\phi + 2\pi k)/n}$$

z = r exp(i theta); z^n = r^n exp(i n theta); the n distinct n-th roots of rho exp(i phi) are rho^(1/n) exp(i(phi + 2 pi k)/n)

The n-th roots of unity form a cyclic group of order n and sum to zero for n at least 2.

Cauchy–Riemann

$$u_x = v_y, \quad u_y = -v_x, \quad f' = u_x + iv_x$$

for f = u + iv: u_x = v_y and u_y = -v_x, with f' = u_x + i v_x; sufficient when the partials are continuous

u and v must both be harmonic, so a non-harmonic real part rules analyticity out immediately.

Cauchy's theorem and integral formula

$$f^{(n)}(a) = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z-a)^{n+1}}\,dz$$

analytic on and inside C gives a zero loop integral; otherwise f(a) = (1/2 pi i) times the loop integral of f(z)/(z-a), and the n-th derivative carries n! with (z-a)^(n+1)

The formula is why analytic functions are infinitely differentiable and why Liouville's theorem holds.

Singularity classification

$$\text{removable} \iff \lim_{z \to a} f \in \mathbb{C}; \quad \text{pole} \iff |f| \to \infty$$

removable when the limit is finite; a pole of order m when the principal part stops at (z-a)^(-m); essential when infinitely many negative powers appear

Near an essential singularity the values come arbitrarily close to every complex number, so no limit exists.

Residues

$$\operatorname{Res}_{z=a}f = \frac{1}{(m-1)!}\lim_{z\to a}\frac{d^{m-1}}{dz^{m-1}}\left[(z-a)^{m}f(z)\right]$$

simple pole: limit of (z-a)f(z), or p(a)/q'(a); pole of order m: the (m-1)-th derivative of (z-a)^m f(z), divided by (m-1)!

Identify the order first; the simple-pole shortcut is wrong at a double pole.

Residue theorem

$$\oint_C f = 2\pi i \sum_k \operatorname{Res}_{z = a_k} f$$

the integral around a positively oriented simple closed contour is 2 pi i times the sum of the enclosed residues

Only singularities strictly inside contribute. Reversing orientation negates the value.

Liouville and maximum modulus

$$f \text{ entire and bounded} \Rightarrow f \text{ constant}$$

a bounded entire function is constant; a nonconstant analytic function attains no interior maximum of its modulus

Neither exp(z) nor sin z is bounded on the complex plane, which is what makes them consistent with Liouville.

Sources

Entries drawn from a cited source list it below. Standard results and numerical constants that no source in our registry covers are given as recall material without a citation rather than attributed to a source that does not support them.

1. GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
2. Calculus Volume 2, Section 5.6: Ratio and Root Tests — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
3. Calculus Volume 2, Section 6.3: Taylor and Maclaurin Series — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
4. MIT OCW 18.06SC Linear Algebra, session: Eigenvalues and Eigenvectors — MIT OpenCourseWare. Accessed 2026-08-03. MIT OpenCourseWare materials are CC BY-NC-SA; attribute MIT OCW, link the source page, and avoid verbatim reuse beyond short cited references.
5. Abstract Algebra: Theory and Applications (Judson), Section 6.2: Lagrange's Theorem — LibreTexts Mathematics. Accessed 2026-08-15. Judson's Abstract Algebra on LibreTexts is released under the GNU Free Documentation License 1.3; attribute the author and platform and avoid verbatim reuse beyond short cited references.
6. Mathematical Analysis (Zakon), Section 4.6: Compact Sets — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.
7. Calculus Volume 3, Section 7.1: Second-Order Linear Equations — OpenStax. Accessed 2026-08-15. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
8. Complex Variables with Applications (Orloff), Section 9.5: Cauchy Residue Theorem — LibreTexts Mathematics. Accessed 2026-08-15. Orloff's Complex Variables with Applications on LibreTexts is CC BY-NC-SA 4.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.