GRE (Graduate Record Examinations) GRE Subject Tests Overview Syllabus

What is the ratio test used for, and what does it conclude?

Testing convergence of a series Σaₙ: if lim |aₙ₊₁/aₙ| = L, the series converges absolutely if L<1, diverges if L>1, and is inconclusive if L=1.

What defines a vector space?

A set with vector addition and scalar multiplication satisfying axioms like associativity, commutativity, a zero vector, additive inverses, and distributivity of scalars.

What does it mean for a set of vectors to be linearly independent?

No vector in the set can be written as a linear combination of the others; equivalently, the only solution to their linear combination equaling zero is all-zero coefficients.

What is an eigenvalue and eigenvector of a matrix A?

A scalar λ and nonzero vector v such that Av = λv; v is stretched/scaled by λ without changing direction.

How is the determinant of a matrix related to invertibility?

A square matrix is invertible if and only if its determinant is nonzero.

In abstract algebra, what four properties define a group?

Closure, associativity, an identity element, and an inverse for each element under a binary operation.

On the GRE Math Subject Test, what does the 'Additional Topics' category roughly cover?

About 25 percent of the test: topics such as discrete mathematics, abstract algebra, number theory, real analysis, topology, probability, statistics, and numerical analysis.

In point-set topology, what is an open set (intuitively)?

A set in which every point has a neighborhood entirely contained in the set; openness is the basic notion used to define continuity and convergence.

What is the definition of a continuous function in terms of epsilon and delta?

f is continuous at a if for every ε>0 there exists δ>0 such that |x−a|<δ implies |f(x)−f(a)|<ε.

What is Bayes' theorem?

P(A|B) = P(B|A)·P(A) / P(B), relating the conditional probability of A given B to that of B given A.

Approximately how much of the GRE Physics Subject Test is devoted to classical mechanics?

About 20 percent, the largest single content area, covering kinematics, Newton's laws, energy, and Lagrangian/Hamiltonian methods.

State Newton's second law of motion.

The net force on an object equals the rate of change of its momentum; for constant mass, F = ma.

What is the Lagrangian of a system, and what equation governs its dynamics?

L = T − V (kinetic minus potential energy); the Euler-Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0 gives the equations of motion.

What do Maxwell's equations describe?

The behavior of electric and magnetic fields: Gauss's law for E, Gauss's law for magnetism, Faraday's law of induction, and the Ampère-Maxwell law.

State Gauss's law for electricity.

The electric flux through a closed surface equals the enclosed charge divided by ε₀: ∮E·dA = Q_enc/ε₀.

What is the Lorentz force on a charged particle?

F = q(E + v × B), the combined force from electric and magnetic fields on a charge q moving with velocity v.

What is the photoelectric effect and what did it demonstrate?

Electrons are ejected from a metal when light above a threshold frequency strikes it; it demonstrated the quantization of light into photons (E = hf).

State the time-independent Schrödinger equation.

Ĥψ = Eψ, where Ĥ is the Hamiltonian operator, ψ the wavefunction, and E the energy eigenvalue.

What is the Heisenberg uncertainty principle for position and momentum?

Δx·Δp ≥ ħ/2; position and momentum cannot both be known with arbitrary precision simultaneously.

In special relativity, what is time dilation?

A moving clock runs slow relative to a stationary observer: Δt = γΔt₀, where γ = 1/√(1−v²/c²).

State the first law of thermodynamics.

The change in internal energy of a system equals heat added to it minus work done by it: ΔU = Q − W (energy conservation).

State the second law of thermodynamics in terms of entropy.

The total entropy of an isolated system never decreases; ΔS ≥ 0, and equals zero only for reversible processes.

What is the relationship between wave speed, frequency, and wavelength?

v = fλ; wave speed equals frequency times wavelength.

What is the Doppler effect for waves?

The observed frequency of a wave changes when the source and observer move relative to each other, increasing as they approach and decreasing as they recede.

In experimental physics, how is the standard error of the mean calculated?

Standard error = s/√n, where s is the sample standard deviation and n is the number of measurements; it quantifies uncertainty in the estimated mean.

What is the difference between accuracy and precision in laboratory measurement?

Accuracy is closeness to the true value; precision is reproducibility (closeness of repeated measurements to each other).

In biological psychology, what is the resting membrane potential of a typical neuron and what causes an action potential?

About −70 mV; an action potential is a rapid depolarization caused by voltage-gated Na⁺ channels opening, followed by repolarization via K⁺ efflux.

In cognitive psychology, what is the distinction between short-term and long-term memory in the modal model?

Short-term memory holds limited information briefly (about 7±2 items for seconds), while long-term memory stores large amounts of information durably after encoding.

What is the difference between classical and operant conditioning?

Classical conditioning associates a neutral stimulus with a reflexive response (Pavlov); operant conditioning changes voluntary behavior through reinforcement or punishment (Skinner).

In social psychology, what is the fundamental attribution error?

The tendency to overemphasize dispositional (personality) causes and underestimate situational causes when explaining others' behavior.

What did Piaget's theory of cognitive development propose as its four stages?

Sensorimotor, preoperational, concrete operational, and formal operational stages.

In clinical psychology and measurement, what is the difference between reliability and validity of a test?

Reliability is the consistency or repeatability of measurement; validity is whether the test actually measures what it claims to measure.