Glossary

Terms used in this book.

A reference list of the mathematical and physical vocabulary used in Math Foundations. Inline occurrences in the chapters are auto-tooltipped.

313 terms from this book.

A

adaptive step size: Automatically adjusting the integration step h based on local error estimates. Smaller steps where the solution changes fast.
aliasing: The artefact that occurs when a signal is sampled below the Nyquist rate: high frequencies masquerade as lower ones, folding back into the baseband.
amplitude: The magnitude of a wave's departure from equilibrium. For sound, the size of the pressure fluctuation.
angular frequency: Rate of phase advance in radians per second: ω = 2πf.
ansatz: An educated guess for the form of a solution, with unknown parameters to be determined by substitution into the equation.
anti-aliasing filter: An analog low-pass filter applied before sampling to remove content above the Nyquist frequency.
antiderivative: A function F whose derivative equals f: F′(x) = f(x). The indefinite integral ∫f(x)dx = F(x) + C.
argument: The phase angle of a complex number: arg(z) = atan2(b, a) for z = a + bi. Represents the phase shift when z is a phasor.
attractor: A set in state space toward which trajectories converge as t → ∞. Can be a fixed point, a closed loop (limit cycle), or a fractal "strange" set.
autocorrelation: The correlation of a signal with a time-shifted copy of itself: R(τ) = ⟨x(t)x(t+τ)⟩. Its Fourier transform is the power spectral density (Wiener-Khinchin).

B

band-limited: A signal containing no frequency content above some maximum frequency B. Such signals can be perfectly reconstructed from samples at rate ≥ 2B.
basis: A minimal spanning set for a vector space: every vector can be written uniquely as a linear combination of basis vectors. An n-dimensional space has exactly n basis vectors.
Bayes: Bayes' theorem: P(M|S) = P(S|M)·P(M)/P(S). The posterior over hypotheses given data combines likelihood with prior.
Bayes' theorem: P(M|S) = P(S|M)·P(M)/P(S). The brain's posterior over hypotheses given sensory data combines the likelihood with the prior.
Bayesian updating: Applying Bayes' rule sequentially: each posterior becomes the next prior. The foundation of online estimation and Kalman filtering.
Bernoulli distribution: Two outcomes: 1 with probability p, 0 with probability 1−p. A single coin toss. Building block of the binomial.
Bessel function: Solutions to Bessel's differential equation, arising in problems with cylindrical symmetry. J_n(x) are finite at the origin; Y_n(x) diverge there.
bifurcation: A qualitative change in a system's long-term behaviour as a parameter crosses a critical value — a fixed point losing stability, or a cycle splitting in two (period-doubling).
binomial distribution: The number of successes in n independent Bernoulli(p) trials. Pr(X=k) = C(n,k)pᵏ(1−p)^{n−k}.
binomial series: The Taylor expansion of (1+x)ᵖ for general p: 1 + px + p(p−1)x²/2! + … Converges for |x| < 1.
bisection method: A root-finding algorithm that repeatedly halves an interval [a,b] where f changes sign. Guaranteed convergence but only linear (one bit per step).
Bode plot: A log-frequency plot of transfer-function magnitude and phase; the standard visualisation of filter response.
boundary-value problem: A differential equation with conditions specified at two or more spatial points, rather than at an initial time.
Brownian motion: The random motion of a particle suspended in a fluid, driven by molecular collisions. Mathematically: a continuous-time stochastic process with Gaussian independent increments.
Buckingham π theorem: If a physical law involves n variables with k independent dimensions, it can be rewritten in terms of n−k dimensionless groups (π groups).
butterfly: The elementary operation of the FFT: combining two partial DFT values with a twiddle factor. The building block of Cooley-Tukey.

C

catastrophic cancellation: Severe loss of significant digits when subtracting two nearly-equal floating-point numbers.
Cauchy–Schwarz inequality: For any vectors u, v in an inner-product space: |⟨u,v⟩| ≤ ‖u‖·‖v‖, with equality iff u and v are parallel. The most-used inequality in functional analysis.
central limit theorem: The sum of many independent random variables (with finite variance) converges to a Gaussian distribution regardless of the individual distributions.
centred difference: The finite-difference approximation [f(x+h)−f(x−h)]/(2h). Second-order accurate: error is O(h²).
CFL condition: The stability bound for explicit finite-difference schemes: the Courant number C = c·Δt/Δx must satisfy C ≤ 1. Numerical signal speed must not exceed physical signal speed.
chain rule: The derivative of a composition: d/dx[f(g(x))] = f′(g(x))·g′(x). The single most-used differentiation technique.
characteristic curve: Curve in (x,t) space along which information propagates; for the wave equation, lines x ± ct = const.
characteristic polynomial: det(A − λI) = 0: a polynomial in λ whose roots are the eigenvalues of A. Degree n for an n×n matrix.
completeness: An orthogonal set is complete if any function can be expanded in it with no leftover residual. Makes Fourier and modal expansions exact.
complex conjugate: For z = a + bi, its conjugate is z* = a − bi. The product z·z* = |z|² is always real and non-negative.
complex plane: The plane with real part horizontal and imaginary part vertical. Phasors, eigenvalues, and impedances all live here.
composition: Applying one function to the output of another: h(x) = f(g(x)). Differentiated by the chain rule.
conjugate prior: A prior distribution that, when combined with a specific likelihood via Bayes' theorem, yields a posterior of the same distributional family. Simplifies sequential updating.
constructive interference: When two waves combine in phase, doubling amplitude. Aligned phasors sum to twice their individual magnitude.
continuity equation: The local conservation law: ∂ρ/∂t + ∇·(ρv) = 0. Any density change equals the negative divergence of the flux.
continuous: A function with no jumps, holes, or breaks: lim_{x→a} f(x) = f(a) for every point a in its domain.
convergence: A sequence or series approaches a definite value as n grows. For a Taylor series, convergence holds within the radius of convergence.
convolution: A mathematical operation combining two signals: the output at time t is the weighted sum of one signal across all times, weighted by the other shifted to t.
convolution theorem: Convolution in the time domain corresponds to multiplication in the frequency domain: F{f*g} = F{f}·F{g}. The computational basis for fast filtering via FFT.
Cooley-Tukey: The divide-and-conquer FFT algorithm that computes a length-N DFT in O(N log N) operations by recursively splitting into even- and odd-indexed sub-transforms.
cosine: For a point at angle θ on the unit circle, its horizontal coordinate, cos θ. Equal to sine shifted by a quarter turn: cos θ = sin(θ + π/2).
Courant number: The dimensionless ratio C = c·Δt/Δx in a finite-difference discretisation of the wave equation. The CFL condition requires C ≤ 1 for stability.
covariance: A measure of joint variability between two random variables: Cov(X,Y) = E[(X−μ_X)(Y−μ_Y)]. Zero for independent variables; sign indicates direction of association.
Crank-Nicolson: A second-order implicit finite-difference scheme for parabolic PDEs that averages the spatial operator between the current and next time step. Unconditionally stable.
critically damped: A damped oscillator with γ = ω₀: returns to equilibrium as fast as possible without oscillating. The design target for many engineered systems.
cumulative distribution: F(x) = P(X ≤ x). The integral of the PDF from −∞ to x. Monotonically non-decreasing from 0 to 1.
curl: A vector measuring local rotation of a field: ∇×v. Its magnitude gives the rotation rate; its direction is the axis of rotation (right-hand rule).

D

d-prime: d′ = (μ_signal − μ_noise)/σ. A bias-free measure of detection sensitivity in signal detection theory. Higher d′ = easier to tell signal from noise.
d'Alembert's solution: The general solution to the 1-D wave equation: u(x,t) = f(x−ct) + g(x+ct). Any disturbance splits into two pulses traveling in opposite directions.
damped natural frequency: The oscillation frequency of a damped system: ω_d = √(ω₀² − γ²), slightly lower than the undamped natural frequency.
damped oscillation: Oscillation with energy loss (friction, viscosity, radiation). Amplitude decays exponentially; the system eventually returns to equilibrium.
damping rate: The coefficient γ controlling exponential energy loss in a damped oscillator. Sets the decay envelope e^{−γt}.
de Moivre's theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). A consequence of Euler's formula; simplifies trigonometric identities and nth-root extraction.
decibel: A logarithmic unit of ratio: 20·log10(amplitude ratio) or 10·log10(power ratio). Used for sound pressure level (SPL) and hearing level (HL).
degenerate: Two or more modes sharing the same eigenfrequency, typically due to domain symmetry (e.g. a square cavity).
del: The vector differential operator ∇ = (∂/∂x, ∂/∂y, ∂/∂z). Applied to a scalar gives gradient; dot with a vector gives divergence; cross gives curl.
derivative: The instantaneous rate of change of a function: f′(x) = lim_{h→0} [f(x+h)−f(x)]/h. Geometrically, the slope of the tangent line.
destructive interference: When two waves combine with opposite phase, reducing or cancelling amplitude. Phasors 180° apart sum to zero.
determinant: A scalar encoding the signed volume scaling of a linear map. det(A) = 0 means the map is singular (collapses a dimension).
deterministic chaos: Long-term unpredictable behaviour produced by a deterministic system with no randomness, arising from sensitive dependence on initial conditions. Bounded, aperiodic, and exponentially sensitive to its starting point.
DFT: Discrete Fourier Transform. Maps N samples in the time domain to N complex coefficients in the frequency domain. Computed efficiently via the FFT in O(N log N).
diagonalisation: Writing a matrix as A = PDP⁻¹ where D is diagonal (eigenvalues) and P holds eigenvectors. Makes computing Aⁿ and e^{At} trivial.
differential equation: An equation relating an unknown function to its derivatives. ODEs involve one variable; PDEs involve several.
differential operator: A linear map that acts on functions by differentiation, e.g. L = d²/dt² + 2γ d/dt + ω₀².
diffusion equation: The parabolic PDE ∂u/∂t = D∇²u describing how a concentration or temperature field spreads out over time. Solutions are Gaussian spreading profiles.
diffusivity: The constant D in the diffusion equation u_t = D∇²u; sets how quickly spatial gradients are smoothed.
dimensionless number: A ratio of physical quantities that has no units. Characterises the relative importance of different physical effects (e.g. Re, Ma, Q).
Dirac delta: A generalised function that is zero everywhere except at the origin, with unit integral: ∫δ(x)dx = 1. The identity element for convolution.
directional derivative: The rate of change of a function along a specified direction û: D_û f = ∇f · û. The gradient gives the maximum directional derivative.
Dirichlet boundary: A boundary condition specifying the value of the solution on the boundary: u|∂Ω = g. Physically: a fixed displacement, a grounded conductor, a clamped edge.
dispersion relation: The relation between frequency ω and wavenumber k for a wave. Non-dispersive: ω = ck. Dispersive: ω(k) is nonlinear.
divergence: A scalar measuring how much a vector field spreads from a point: ∇·v = ∂vₓ/∂x + ∂vᵧ/∂y + ∂v_z/∂z. Positive = source; negative = sink.
divergence theorem: Relates a volume integral of a divergence to a surface integral: ∫∫∫(∇·F)dV = ∮(F·n̂)dA. Converts between flux through a surface and source strength inside.
domain of dependence: The set of initial-data points that can influence the solution at a given (x,t). For the wave equation: the backward characteristic cone.
Dormand-Prince: An adaptive-step embedded Runge-Kutta pair (orders 4 and 5) that estimates local error by comparing two solutions. The default integrator in most scientific computing libraries.
dot product: The sum of componentwise products: u·v = Σuₖvₖ. Gives lengths, angles, and projections of vectors.

E

eigenfunction: A function that a linear operator merely multiplies by a scalar: L[f] = λf. Modes of vibration are eigenfunctions of the wave operator.
eigenmode: A natural vibration pattern of a bounded system, corresponding to a specific eigenfrequency. Each eigenmode shape satisfies the boundary conditions.
eigenstate: An eigenfunction of a linear operator, interpreted as a state of the system. Energy eigenstates form a complete basis for any wavefunction.
eigenvalue: A scalar λ such that Av = λv for some nonzero vector v. The eigenvalues of a matrix determine its stretching/shrinking along principal directions.
eigenvector: A nonzero vector v satisfying Av = λv. Points along a direction that the linear map A merely scales (by λ) without rotating.
elliptic PDE: A PDE with no time evolution (like Laplace's equation). Boundary data determines the interior solution globally.
equilibrium: A state where all rates of change are zero; the system remains there unless disturbed.
Euler's formula: e^{iθ} = cos θ + i sin θ. Connects the exponential function to trigonometry; the foundation of phasor analysis and Fourier theory.
expectation: The probability-weighted average of a random variable: E[X] = ∫x·f(x)dx (continuous) or Σx·P(x) (discrete). Also called the mean or first moment.
explicit method: A numerical scheme computing the next value directly from known current values. Simple but stability-limited.
exponential: The function eˣ, the unique function equal to its own derivative. Basis of growth, decay, oscillation (via Euler's formula), and phasor analysis.
exponential decay: The solution x(t) = x₀e^{−αt} of dx/dt = −αx. The generic loss process: decay rate proportional to what remains.
exponential distribution: f(t) = λe^{−λt} for t ≥ 0. The distribution of inter-arrival times in a Poisson process. Memoryless: P(T > s+t | T > s) = P(T > t).
exponential function: The function eˣ, the unique function equal to its own derivative. Solution to dx/dt = x. Basis of growth, decay, and phasor analysis.

F

false alarm rate: The probability of reporting 'signal present' when only noise is present; one axis of the ROC curve.
Feigenbaum constant: The universal ratio δ ≈ 4.6692 at which successive period-doubling bifurcations accumulate, identical for every smooth unimodal map. Evidence that the route to chaos is universal.
FFT: Fast Fourier Transform. An O(N log N) algorithm for computing the DFT, versus the naive O(N²). Enables real-time spectral analysis.
finite difference: Approximation of a derivative by a difference quotient on a grid: f′(x) ≈ [f(x+h)−f(x−h)]/(2h) (centred, second-order).
fixed point: A value x* such that f(x*) = x*. Iterative methods converge when the mapping is contractive near the fixed point (|f′(x*)| < 1).
flux: The flow of a vector field through a surface: ∫∫F·dA. Measures how much of the field passes through the area.
forcing: An external drive applied to a system. The particular solution responds to forcing; the homogeneous solution decays away.
forward Euler: The simplest ODE integrator: y_{n+1} = y_n + h·f(t_n, y_n). First-order accurate; conditionally stable. Useful pedagogically but rarely in production.
Fourier coefficient: The complex amplitude cₙ of the n-th harmonic in a Fourier series. Extracted by projection: cₙ = (1/T)∫f(t)e^{−inω₀t}dt.
Fourier series: Decomposition of a periodic signal into a sum of sinusoids at multiples of its fundamental frequency.
Fourier transform: A mathematical operation that decomposes a signal into its sinusoidal components. Time-domain ↔ frequency-domain pair.
frequency: The number of oscillation cycles per second, measured in hertz (Hz). For sound, this is what the brain perceives as pitch.
fundamental frequency: The lowest frequency in a periodic signal's Fourier series: f₁ = 1/T where T is the period. All other harmonics are integer multiples.
fundamental theorem of calculus: Links differentiation and integration: d/dx ∫ₐˣ f(t)dt = f(x), and ∫ₐᵇ f(x)dx = F(b)−F(a) where F′ = f.
FWHM: Full Width at Half Maximum — the frequency (or other) interval where a peak exceeds half its peak value.

G

Gauss-Seidel: An iterative method for solving Ax = b that updates each variable using the latest available values of other variables. Converges for diagonally-dominant or symmetric positive-definite A.
Gaussian distribution: The normal distribution: f(x) = (1/σ√2π)exp(−(x−μ)²/2σ²). The Central Limit Theorem makes it ubiquitous for sums of independent random variables.
Gaussian elimination: The standard algorithm for solving Ax = b: reduce to upper-triangular form by row operations, then back-substitute.
general solution: The full family of solutions to a differential equation, containing as many free constants as the order.
Gibbs phenomenon: The ~9% overshoot that persists at a discontinuity when a function is represented by a truncated Fourier series, no matter how many terms are included.
gradient: The vector of partial derivatives ∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z). Points in the direction of steepest increase; its magnitude is the rate of increase.
Gram-Schmidt: An algorithm that converts any linearly independent set into an orthonormal basis by sequential projection and normalisation.
Green's function: The response of a linear system to a unit impulse. Fully characterises propagation from source to receiver in any linear medium.

H

half-life: The time for a decaying quantity to reach half its initial value; related to time constant by T½ = τ ln 2.
Hamiltonian: The operator Ĥ = −(ℏ²/2m)∇² + V representing total energy; its eigenvalues are the allowed energies.
Hann window: A raised-cosine window w(n) = ½[1 − cos(2πn/N)] used to reduce spectral leakage in DFT analysis.
harmonic: An integer multiple of the fundamental frequency. The nth harmonic has frequency nf₁. Harmonics are the building blocks of periodic signals.
harmonic function: A function satisfying Laplace's equation ∇²u = 0. Its value at any interior point equals the average on a surrounding sphere (the mean-value property).
heat equation: The parabolic PDE ∂u/∂t = α∇²u describing thermal conduction (or any diffusive process). Solutions smooth out discontinuities instantly.
Helmholtz decomposition: Any vector field splits into an irrotational part (gradient of a scalar) plus a solenoidal part (curl of a vector).
Helmholtz equation: The time-independent wave equation: ∇²u + k²u = 0. Arises from separating the time dependence out of the wave equation for steady-state oscillations.
Hermitian: A complex matrix or operator equal to its own conjugate transpose (A† = A). Hermitian operators have real eigenvalues and orthogonal eigenvectors forming a complete basis.
Hilbert space: A complete inner-product space — the infinite-dimensional generalisation of Euclidean space. The natural setting for Fourier analysis, quantum mechanics, and PDE solutions.
homogeneous: A differential equation with zero forcing (RHS = 0). Solutions obey superposition and can be freely added.
hyperbolic PDE: A PDE with finite propagation speed (like the wave equation). Information travels along characteristics.

I

i.i.d.: Independent and identically distributed. Random variables that share the same distribution and carry no information about each other. The standard simplifying assumption.
identity matrix: The square matrix I with 1s on the diagonal and 0s elsewhere. The "do nothing" transformation: Iv = v for all v.
imaginary unit: The number i satisfying i² = −1. Extends the reals to the complex plane and enables phasor algebra.
implicit method: A numerical scheme whose update depends on the (unknown) future value. Requires solving an equation at each step but is unconditionally stable.
impulse response: A system's output when given a single, brief impulse as input. Fully characterises any linear time-invariant system.
initial condition: The value(s) of the solution and its derivatives at the starting time: y(0) = y₀, y′(0) = v₀. Picks out a unique solution from the general family.
inner product: A generalisation of the dot product to abstract vector spaces: ⟨f, g⟩ = ∫f*(x)g(x)dx for functions. Defines orthogonality, norms, and projections.
integral: The signed area under a curve: ∫f(x)dx. The Fundamental Theorem connects it to the antiderivative. Extends to line, surface, and volume integrals.
integration by parts: ∫u dv = uv − ∫v du. Derived from the product rule; converts one integral into another (hopefully simpler) one.
inter-arrival time: The random waiting time between consecutive events in a Poisson process; exponentially distributed with mean 1/λ.
inverse Fourier transform: Recovers a time-domain signal from its frequency-domain representation: f(t) = (1/2π)∫F(ω)e^{iωt}dω. The inverse of the forward transform.
inverse function: A function f⁻¹ that undoes f: f⁻¹(f(x)) = x. Exists when f is one-to-one. Its graph is the reflection of f across y = x.
inverse rule: If y = f(x) and x = f⁻¹(y), then dx/dy = 1/(dy/dx). The slope of an inverse function is the reciprocal of the original.
irreversible: A process that cannot be undone by reversing time; the heat equation loses information as high-k modes decay.
irrotational: A vector field with zero curl everywhere (∇×v = 0). Such a field can be written as the gradient of a scalar potential: v = ∇φ.

J

Jacobi iteration: The simplest relaxation method: replace each interior grid value by the average of its neighbours. Trivially parallel but slow to converge.
Jacobian: The matrix of all first-order partial derivatives of a vector-valued function. Locally approximates a nonlinear map as a linear one; its determinant gives the volume scaling factor.

L

L'Hôpital's rule: For indeterminate limits 0/0 or ∞/∞: lim f/g = lim f'/g' when the latter exists. Resolves ambiguous limiting ratios.
Laplace equation: The elliptic PDE ∇²φ = 0. Governs equilibrium fields: steady-state temperature, electrostatic potential, incompressible potential flow.
Laplace transform: An integral transform F(s) = ∫₀^∞ f(t)e^{−st} dt that converts ODEs with initial conditions into algebraic equations. The s-domain generalises the Fourier domain to complex frequencies.
Laplacian: The divergence of the gradient: ∇²φ = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z². Measures how a function deviates from its local average.
law of large numbers: The sample mean converges to the true mean as sample size grows: (1/n)Σxᵢ → E[X]. The mathematical basis for empirical measurement.
leapfrog scheme: A time-centred finite-difference scheme: y_{n+1} = y_{n−1} + 2h·f(t_n, y_n). Second-order accurate and energy-conserving for Hamiltonian systems.
least-squares: Finding x that minimises ‖Ax − b‖² for an overdetermined system; solved via the normal equations AᵀAx = Aᵀb.
Legendre polynomial: Orthogonal polynomial P_ℓ(cos θ) solving the angular part of Laplace's equation in spherical coordinates.
level set: The set of points where a scalar field takes a constant value; contour lines in 2-D, isosurfaces in 3-D.
likelihood: In Bayesian inference, P(S|M): how probable the observed sensory input is given that hypothesis M is true.
likelihood ratio: The ratio f₁(x)/f₀(x) of probability densities under two hypotheses. The optimal statistic for distinguishing signal from noise.
limit: The value a function approaches as its input approaches some point. The foundation of derivatives and integrals.
limit cycle: An isolated closed trajectory in state space toward which neighbouring trajectories spiral. Represents self-sustained periodic oscillation in a nonlinear system.
linear combination: A sum of vectors each multiplied by a scalar: c₁v₁ + c₂v₂ + … Any vector is a linear combination of basis vectors.
linear independence: Vectors {v₁,…,vₙ} are linearly independent if no nontrivial combination c₁v₁+…+cₙvₙ = 0. Independent vectors span an n-dimensional subspace.
linear time-invariant: A system whose output obeys superposition (linear) and whose behaviour does not change over time (time-invariant). Sinusoids are its eigenfunctions.
linearisation: Replacing a function by its first-order Taylor approximation near a point: f(x) ≈ f(x₀) + f′(x₀)(x−x₀). Valid when |x−x₀| is small.
linearity: The property that an operation distributes over addition and commutes with scalar multiplication: L(af + bg) = aL(f) + bL(g).
logarithm: The inverse of the exponential: ln(y) = x means eˣ = y. Its derivative is 1/x, making it the antiderivative of 1/x.
logistic map: The quadratic recurrence x_{n+1} = r x_n (1 − x_n), the simplest system exhibiting the full period-doubling route to chaos as r increases toward 4.
Lorentzian: The function 1/[(ω−ω₀)² + Γ²]; the spectral shape of a damped resonance, universal near any isolated pole.
LTI system: Linear Time-Invariant system. Fully characterised by its impulse response; output = input * h(t) (convolution).
Lyapunov exponent: The average exponential rate λ at which nearby trajectories separate: |Δ(t)| ≈ |Δ(0)| e^{λt}. A positive largest Lyapunov exponent is the quantitative signature of chaos.

M

Maclaurin series: A Taylor series expanded about x = 0. The standard expansions of eˣ, sin x, cos x are all Maclaurin series.
marginal likelihood: The total probability of the data P(D) = Σ P(D|H)P(H); normalises the posterior in Bayes' theorem.
matrix: A rectangular array of numbers representing a linear transformation. An m×n matrix maps n-vectors to m-vectors.
matrix inverse: A⁻¹ satisfying A·A⁻¹ = I. Exists iff det(A) ≠ 0. Solving Ax = b becomes x = A⁻¹b (though direct inversion is rarely the best numerical method).
maximum principle: A harmonic function on a bounded region attains its maximum and minimum on the boundary, never in the interior (unless constant).
method of characteristics: A PDE-solving technique that reduces the PDE to ODEs along curves (characteristics) in the (x,t) plane. Natural for hyperbolic equations.
mode shape: The spatial pattern X_n(x) of a single standing-wave mode; a sinusoidal eigenfunction of the spatial operator.
modulus: The magnitude of a complex number: |z| = √(a² + b²) for z = a + bi. Represents the amplitude when z is a phasor.
Monte Carlo: A class of algorithms that use random sampling to estimate numerical quantities — integrals, expectations, probabilities — that are hard to compute deterministically.
multigrid: A numerical method that accelerates convergence of iterative solvers by solving the residual equation on successively coarser grids, then interpolating corrections back up.

N

natural frequency: The frequency at which an undamped system oscillates freely: ω₀ = √(k/m) for a mass-spring. The intrinsic resonance of the system.
natural logarithm: The logarithm to base e, written ln x. Inverse of eˣ, and the base in which log laws carry no extra constant (d/dx ln x = 1/x).
Neumann boundary: A boundary condition specifying the normal derivative on the boundary: ∂u/∂n|∂Ω = h. Physically: a specified flux, a free end, a perfectly reflecting wall.
Newton's method: An iterative root-finding algorithm: x_{n+1} = x_n − f(x_n)/f′(x_n). Converges quadratically near a simple root when started sufficiently close.
nodal line: A curve on a vibrating surface where the displacement is always zero; separates regions oscillating in antiphase.
norm: The length of a vector: ‖v‖ = √(v·v). Generalises Pythagoras to n dimensions.
normal mode: A pattern of motion in which all parts of a system oscillate at the same frequency and in a fixed phase relationship. The building blocks of any vibration.
Nyquist frequency: Half the sample rate: fₛ/2. The maximum frequency representable without aliasing in a sampled signal.
Nyquist plot: A parametric plot of a complex transfer function in the complex plane as frequency sweeps; used for stability analysis.
Nyquist rate: The minimum sampling rate (2× the highest frequency present) required to perfectly reconstruct a band-limited signal from its samples.

O

optimisation: Finding the input that maximises or minimises a function. At an optimum, the derivative is zero (necessary condition for smooth functions).
order of accuracy: The power of h in the leading truncation-error term. A p-th order method has error O(hᵖ); halving h divides error by 2ᵖ.
ordinary differential equation: An equation relating a function to its derivatives with respect to a single independent variable. Order = highest derivative present.
orthogonality: Two vectors are orthogonal when their inner product is zero: ⟨u,v⟩ = 0. Generalises perpendicularity to abstract spaces; the basis of Fourier decomposition.
orthonormal basis: A basis whose vectors are mutually perpendicular and unit-length: ⟨eᵢ, eⱼ⟩ = δᵢⱼ. Makes projections trivial: component = ⟨v, eᵢ⟩.
overdamped: A damped oscillator with γ > ω₀: returns to equilibrium without oscillating but slower than critical damping. Two real decay rates.

P

parabolic PDE: A PDE with diffusive character (like the heat equation). Disturbances smooth out over time at all speeds.
Parseval's theorem: Total energy in the time domain equals total energy in the frequency domain: ∫|f(t)|²dt = ∫|F(ω)|²dω. Energy is preserved by the Fourier transform.
partial derivative: The derivative of a multivariable function with respect to one variable, holding the others fixed: ∂f/∂x.
partial differential equation: An equation involving partial derivatives of an unknown function of two or more variables. Classification: elliptic, parabolic, or hyperbolic.
partial fractions: Decomposing a rational function into simpler fractions that each integrate to logarithms or arctangents.
particular solution: A single solution of a non-homogeneous equation matching the forcing. Added to the homogeneous solution for the full answer.
period-doubling: A bifurcation in which a stable cycle of period T is replaced by a stable cycle of period 2T. An infinite cascade of period-doublings, accumulating geometrically, is one universal route to chaos.
phase portrait: A plot of trajectories in the (position, velocity) plane for a dynamical system. Reveals fixed points, limit cycles, and qualitative behaviour at a glance.
phasor: A complex number Ae^{iφ} representing a sinusoidal signal's amplitude A and phase φ. Converts differential equations into algebraic ones by replacing d/dt with iω.
pivot: The diagonal entry used to eliminate entries below it in Gaussian elimination; must be non-zero.
Plancherel theorem: The Fourier transform preserves the L² inner product (up to a 2π factor); generalises Parseval's identity.
Planck constant: Fundamental quantum of action ℏ ≈ 1.055×10⁻³⁴ J·s; sets the scale at which quantum effects become significant.
Poisson distribution: Distribution of event counts: P(k) = (λT)^k e^{−λT}/k!, with mean = variance = λT.
Poisson process: A stochastic process in which events occur independently at a constant average rate λ. The number of events in any interval is Poisson-distributed with mean λt.
polar form: Writing a complex number as z = re^{iθ} with modulus r = |z| and argument θ. Multiplication becomes scaling and rotation.
polynomial: A function built from powers of x with constant coefficients: aₙxⁿ + … + a₁x + a₀. Degree n means the highest power is n.
positive definite: A matrix A for which vᵀAv > 0 for all nonzero v. Ensures a quadratic form has a unique minimum.
posterior: In Bayesian inference, P(M|S): the probability of hypothesis M given sensory input S. What the brain perceives.
power spectral density: The squared magnitude of a signal's Fourier transform per unit time; gives power per unit frequency.
power spectrum: The distribution of signal power across frequency: |F(ω)|². Its integral equals the total signal energy (Parseval's theorem).
precision: The reciprocal of variance (1/σ²); in Bayesian updating, precisions add when combining prior and data.
prior: In Bayesian inference, P(M): the probability of a hypothesis before any sensory input. Shaped by context, expectation, and learned experience.
probability density function: A function f(x) ≥ 0 whose integral over any interval gives the probability of the random variable falling in that interval. ∫f(x)dx = 1 over the full domain.
product rule: The derivative of a product: (fg)' = f'g + fg'. Each factor is differentiated in turn while the other holds still.
projection: The component of one vector along another: proj_u(v) = (⟨v,u⟩/⟨u,u⟩)u. Fourier coefficients are projections onto sinusoidal basis functions.

Q

quadratic convergence: Error at each step is proportional to the previous error squared. Digits of accuracy double per iteration. Newton's method achieves this near a simple root.
quantisation condition: The constraint (e.g. kL = nπ) that selects a discrete set of allowed wavenumbers from boundary conditions.
quotient rule: The derivative of a ratio: (f/g)' = (f'g − fg')/g². A consequence of the product and chain rules.

R

radian: The natural measure of angle: the arc length subtended on a circle of radius 1. A full turn is 2π radians (360°). Radians are the unit in which d/dθ sin θ = cos θ.
radius of convergence: The distance from the expansion point within which a power series converges absolutely. Beyond it, the series diverges.
radix-2: FFT variant requiring N to be a power of 2; splits the DFT into even- and odd-indexed halves at each level.
random variable: A quantity determined by a random process. Described by its distribution (PMF for discrete, PDF for continuous).
random walk: A stochastic process consisting of successive random steps. In the simplest form: ±1 steps with equal probability. The mean displacement grows as √N.
rate of change: How quickly a quantity varies with respect to another variable. The derivative gives the instantaneous rate of change.
Rayleigh quotient: For a matrix A and vector v: R(v) = vᵀAv/vᵀv. Bounded by the eigenvalues of A; iterating on power-method iterates converges to the dominant eigenvalue.
reactance: The imaginary part X of a complex impedance Z = R + iX; stores energy without dissipating it.
region of influence: The set of spacetime points that can be affected by a given initial-data point; forward-opening wedge bounded by characteristics.
restoring force: Force directed toward equilibrium, proportional to displacement for small perturbations (F = −kx).
Riemann sum: An approximation of a definite integral as a finite sum of rectangle areas. Converges to the integral as the partition refines.
Robin boundary: A boundary condition that is a linear combination of Dirichlet and Neumann: αu + β∂u/∂n = g. Models imperfect absorption or impedance boundaries.
ROC curve: Receiver Operating Characteristic: a plot of true-positive rate vs. false-positive rate as the detection threshold varies. Area under the curve measures discriminability.
root-mean-square: The square root of the time-averaged square of a signal: f_rms = √((1/T)∫f²dt). For a sinusoid, amplitude/√2.
Runge-Kutta: A family of single-step ODE solvers that evaluate the derivative at multiple trial points per step to achieve high-order accuracy. RK4 (the "classic" method) is fourth-order.

S

sample rate: Samples per second f_s when digitising a signal; must exceed 2× the maximum frequency to avoid aliasing.
sampling theorem: Shannon's theorem: a band-limited signal with maximum frequency B is completely determined by samples taken at rate ≥ 2B (the Nyquist rate).
scalar field: A function assigning a single number to each point in space (temperature, pressure, potential). Visualised as contour maps.
scaling law: A power-law relationship between quantities that holds across scales. Often derivable from dimensional analysis without solving the full equations.
secant method: A root-finding algorithm like Newton's method but replacing the derivative with a finite-difference approximation from the two most recent iterates. Superlinear convergence (order ≈1.618).
self-adjoint: A linear operator that equals its own adjoint. For real matrices: A = Aᵀ (symmetric). For differential operators: ⟨f, Lg⟩ = ⟨Lf, g⟩. Guarantees real eigenvalues and orthogonal eigenfunctions.
sensitive dependence on initial conditions: The hallmark of chaos: two trajectories that start arbitrarily close together separate exponentially fast, at a rate set by the largest Lyapunov exponent. Small uncertainties are amplified until prediction fails.
separation constant: The shared constant that both sides of a separated PDE must equal; becomes the eigenvalue of the spatial problem.
separation of variables: A PDE-solving technique that assumes the solution factors as u(x,t) = X(x)·T(t), reducing one PDE to two ODEs. Works when the operator and domain allow it.
signal detection theory: A framework for analysing decisions under uncertainty: the observer must decide whether a noisy observation came from a signal-present or signal-absent distribution.
simple harmonic motion: Oscillation governed by ẍ = −ω₀²x. Solution: x(t) = A cos(ω₀t + φ). The linearised dynamics near any stable equilibrium.
sinc function: sinc(x) = sin(πx)/(πx). The Fourier transform of a rectangular pulse and the ideal interpolation kernel for band-limited reconstruction.
sine: For a point at angle θ on the unit circle, its vertical coordinate, sin θ. Periodic with period 2π; the building block of oscillations and Fourier series.
singular matrix: A square matrix with determinant zero. Not invertible; its columns are linearly dependent.
sinusoid: A function of the form A sin(ωt + φ) or equivalently A cos(ωt + φ). The basic periodic waveform; all Fourier components are sinusoids.
slope: The ratio of vertical to horizontal change along a line or curve. For a curve, the slope at a point is the derivative there.
slope field: A plot showing the local slope f(t,x) at every point; solutions are curves threading tangentially through the field.
solenoidal: A vector field with zero divergence everywhere (∇·v = 0). Incompressible fluid flow is solenoidal; such fields can be written as the curl of a vector potential.
Sommerfeld radiation condition: The boundary condition at infinity for an unbounded Helmholtz problem: φ at large r must look like an outgoing spherical wave (~e^{ikr}/r), not incoming.
sparse matrix: A matrix with mostly zero entries; the 5-point Laplacian stencil on an N×N grid has only 5 non-zeros per row.
spectral leakage: Energy spreading from a signal's true frequency into adjacent DFT bins when the analysis window does not contain an integer number of cycles. Mitigated by window functions.
spectral method: A PDE-solving technique that expands the solution in a global basis (Fourier, Chebyshev) and solves for the coefficients. Exponential convergence for smooth problems.
spectral radius: The largest absolute eigenvalue of an iteration matrix; must be < 1 for convergence of iterative methods.
spectral theorem: A self-adjoint linear operator has a complete orthonormal basis of eigenvectors with real eigenvalues. The reason mode expansions and separation of variables work.
spectrum: The frequency-domain representation of a signal: the set of amplitudes and phases at each frequency component.
spherical harmonic: Eigenfunction Y_ℓm(θ,φ) of the angular Laplacian on the sphere; the angular modes in spherical geometry.
stable spiral: A fixed point with complex eigenvalues having negative real part; trajectories spiral inward.
standard deviation: The square root of variance: σ = √Var(X). Measures spread in the same units as the variable. ≈68% of a Gaussian falls within ±1σ of the mean.
steady state: The long-time behaviour of a driven system after transients have died away. For a sinusoidally driven linear system, it oscillates at the driving frequency.
stencil: The pattern of grid points used in a finite-difference formula. The 5-point stencil approximates the 2-D Laplacian.
stiff system: An ODE system with widely separated time scales. Explicit methods are forced into tiny steps; implicit methods are required.
stochastic process: A family of random variables indexed by time or space. Brownian motion and Poisson processes are the two canonical examples.
Stokes theorem: Relates a surface integral of curl to a line integral around the boundary: ∫∫(∇×F)·dA = ∮F·dr. Generalises Green's theorem to 3-D.
strange attractor: An attractor with fractal (non-integer) dimension on which the motion is chaotic. Trajectories are confined to it yet never repeat and never cross — the Lorenz butterfly is the canonical example.
Strouhal number: Dimensionless St = fd/U relating vortex-shedding frequency to flow speed and body size; St ≈ 0.2.
Sturm–Liouville theorem: For a self-adjoint differential operator on a bounded interval with appropriate boundary conditions, there is a countably infinite sequence of real eigenvalues and a complete orthonormal basis of eigenfunctions.
substitution: An integration technique mirroring the chain rule: replace a composite integrand with a simpler variable u = g(x).
successive over-relaxation: An acceleration of Gauss-Seidel using an over-relaxation factor ω > 1. Optimal ω reduces iteration count from O(N²) to O(N).
superposition: The principle that solutions of a linear equation can be added to give new solutions. The foundation of Fourier methods and modal analysis.
symmetric matrix: A matrix equal to its transpose: A = Aᵀ. The spectral theorem guarantees real eigenvalues and orthogonal eigenvectors.

T

tangent: The ratio tan θ = sin θ / cos θ, the slope of the radius to the point at angle θ on the unit circle. Diverges where cos θ = 0.
tangent line: The straight line that touches a curve at a single point and matches its slope there. Its slope equals the derivative at that point.
Taylor expansion: An expansion of a smooth function f near a base point x0 as an infinite sum f(x0) + f′(x0)(x−x0) + ½f″(x0)(x−x0)² + … . Truncating gives a polynomial approximation of any required order.
Taylor polynomial: A polynomial of degree N matching a function and its first N derivatives at a base point. The finite truncation of a Taylor series.
Taylor series: An expansion of a smooth function f near a base point x0 as an infinite sum f(x0) + f′(x0)(x−x0) + ½f″(x0)(x−x0)² + … . Truncating gives a polynomial approximation of any required order.
time constant: τ = 1/α: the time for an exponentially decaying quantity to fall to 1/e ≈ 37% of its initial value.
time-harmonic: Oscillating at a single frequency; a field of the form φ(r)e^{−iωt}, reducing the wave equation to Helmholtz.
total differential: The first-order change in a multivariate function: df = (∂f/∂x)dx + (∂f/∂y)dy + … Measures response to simultaneous small changes.
trace: The sum of diagonal elements of a matrix: tr(A) = Σaᵢᵢ. Equals the sum of eigenvalues. Invariant under change of basis.
transfer function: The frequency-domain ratio of output to input phasor: H(ω) = Y(ω)/X(ω). Fully characterises a linear system's frequency response.
transient: The component of a system's response that decays over time after a perturbation, leaving only the steady state.
transpose: The matrix Aᵀ formed by swapping rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. A symmetric matrix satisfies Aᵀ = A.
truncation error: The error introduced by replacing a continuous operator with a discrete approximation. For a p-th order method: error ∝ hᵖ as h → 0.
twiddle factor: The complex exponential weight W_N^k = e^{−2πik/N} appearing in the DFT. Pre-computing twiddle factors is a key optimisation in FFT implementations.

U

uncertainty principle: A signal cannot be simultaneously narrow in time and narrow in frequency. Time-bandwidth product has a minimum, saturated by the Gaussian.
underdamped: A damped oscillator with γ < ω₀: oscillates with exponentially decaying amplitude. The most common regime for musical and biological systems.
uniform distribution: Constant probability density on an interval [a,b]. The default when only bounds are known and no value is preferred.
unit circle: The circle |z| = 1 in the complex plane. e^{iθ} traces it as θ varies. Phasors of unit amplitude live on it.
unitary: A transformation preserving inner products (and norms); time evolution under the Schrödinger equation is unitary.
unstable node: A fixed point with two real positive eigenvalues; trajectories diverge from it monotonically.

V

variance: The expected squared deviation from the mean: Var(X) = E[(X−μ)²]. Measures the spread of a distribution; its square root is the standard deviation.
vector: An ordered tuple of numbers representing magnitude and direction. The fundamental object of linear algebra.
vector field: A function assigning a vector to each point in space (velocity, electric field, force). Visualised as arrow fields or streamlines.
velocity potential: A scalar field φ such that v = ∇φ. Exists when the flow is irrotational. Central to linearised acoustics.
von Neumann stability analysis: Substituting a Fourier mode into a finite-difference scheme and checking whether its amplitude grows. The standard stability test for PDE schemes.

W

wave equation: A second-order PDE describing how a disturbance propagates. For pressure in air: ∂²p/∂t² = c²∇²p.
wavefunction: Complex-valued function Ψ(r,t) in quantum mechanics whose squared magnitude gives the probability density of position.
wavenumber: The spatial frequency of a wave: k = 2π/λ. Higher k means shorter wavelength and more rapid spatial oscillation.
wavevector: Vector k pointing in propagation direction with magnitude 2π/λ; encodes spatial periodicity and direction.
well-posed problem: A PDE problem (in Hadamard's sense) where a solution exists, is unique, and depends continuously on the data. Ill-posed problems require regularisation.
Wiener process: The continuous-time limit of a random walk: a Gaussian process with independent increments, continuous paths, and variance growing linearly with time. The mathematical model of Brownian motion.
window function: A tapering function (Hann, Hamming, Blackman, etc.) multiplied with a signal segment before DFT to reduce spectral leakage at the cost of frequency resolution.
WKB approximation: A method for solving wave equations with slowly-varying coefficients. Gives A(x)·exp(i·∫κ(x)dx) for position-dependent wavenumber κ.