Mathematical & Computational Physics — 60 Terms Defined

Algorithm Mathematical & Computational Physics Computational Logic A finite, unambiguous sequence of mathematical instructions implemented in software to solve specific physical equations or compute numerical values. Step-by-step computational procedure

Bessel Functions Mathematical & Computational Physics Cylindrical Special Functions A family of canonical special functions J_α(x) and Y_α(x) serving as solutions to Bessel's differential equation. x² y'' + x y' + (x² - α²) y = 0 ⇒ y(x) = A J_α(x) + B Y_α(x)

Big O Notation Mathematical & Computational Physics Algorithmic Asymptotic Complexity A mathematical notation classifying algorithms according to how their execution time or memory requirements grow as input size N increases. f(N) = O(g(N)) if |f(N)| ≤ C |g(N)| for N ≥ N₀

Boundary Value Problem Mathematical & Computational Physics Differential Equation Condition A differential equation together with a set of additional constraints, called boundary conditions, specified at the boundaries of the domain. Dirichlet: y(a)=y_A; Neumann: y'(a)=y'_A; Robin: α y(a) + β y'(a) = γ

Canonical Transformation Mathematical & Computational Physics Phase Space Coordinate Transformation A change of coordinates in phase space (q, p) → (Q, P) that preserves the canonical form of Hamilton's equations. (q, p) → (Q, P) such that Q̇_i = ∂K / ∂P_i, Ṗ_i = -∂K / ∂Q_i

Chaos Theory Mathematical & Computational Physics Divergence Rate λ (s⁻¹) The study of deterministic non-linear dynamical systems that exhibit extreme sensitivity to initial conditions (the butterfly effect). Lyapunov Exponent: δx(t) ≈ δx₀ e^(λ t), λ > 0

Complex Analysis Mathematical & Computational Physics Complex Function Theory The branch of mathematical analysis studying functions of complex variables f(z) = u(x,y) + i v(x,y). ∮_C f(z) dz = 2 π i ∑ Res(f, z_k) (Residue Theorem)

Curl Mathematical & Computational Physics [Vector Field Unit] / m A vector differential operator measuring the local circulation or rotation density of a 3D vector field around a point. curl F = ∇ × F = det|[î, ĵ, k̂], [∂/∂x, ∂/∂y, ∂/∂z], [F_x, F_y, F_z]|

Curve Fitting Mathematical & Computational Physics Goodness of Fit χ² The mathematical construction of a function that best matches a set of experimental data points according to statistical criteria. Minimizing χ² = ∑ ((y_i - f(x_i, θ)) / σ_i)²

Differential Equation Mathematical & Computational Physics Mathematical Equation Class A mathematical equation relating an unknown function to its derivatives with respect to one or more independent variables. F(x, y, y', y'', ..., y^(n)) = 0, d²y/dx² + ω² y = 0

Dirac Delta Function Mathematical & Computational Physics 1 / [x Unit] A generalized function (distribution) that is zero everywhere except at x = 0 where it is infinite, with an integral over all space equal to 1. δ(x) = 0 for x ≠ 0, ∫_(-∞)^(∞) δ(x) dx = 1, ∫ f(x) δ(x - a) dx = f(a)

Divergence Mathematical & Computational Physics [Vector Field Unit] / m A scalar differential operator measuring the net outward flux density of a vector field from an infinitesimal volume around a given point. div F = ∇ · F = (∂F_x/∂x) + (∂F_y/∂y) + (∂F_z/∂z)

Divergence Theorem Mathematical & Computational Physics Vector Integral Theorem A fundamental theorem of vector calculus stating that the outward flux of a vector field through a closed surface ∂V equals the volume integral of its divergence over the enclosed region V. ∯_(∂V) F · dA = ∭_V (∇ · F) dV

Eigenvalue Problem Mathematical & Computational Physics Eigenvalues λ (Scalar Units) The linear algebra problem of finding non-zero eigenvectors v that, when operated on by a linear operator Â, yield a scalar multiple λ of themselves. Â v = λ v, det(A - λ I) = 0

Eigenvalues and Eigenvectors Mathematical & Computational Physics Linear Algebra Object Special vectors v associated with a linear transformation matrix A that maintain their spatial direction when operated on, scaled only by factor λ. A v = λ v (where v ≠ 0)

Error Propagation Mathematical & Computational Physics Uncertainty Units The mathematical evaluation of how uncertainties in measured input variables propagate into the final calculated physical quantity. σ_f² = ∑ (∂f / ∂x_i)² σ_x_i² + 2 ∑ (∂f/∂x_i)(∂f/∂x_j) Cov(x_i, x_j)

Euler Method Mathematical & Computational Physics First-Order Numerical ODE Integration A basic first-order numerical procedure for solving ordinary differential equations with a given initial value. y_(n+1) = y_n + h f(t_n, y_n)

Euler-Lagrange Equation Mathematical & Computational Physics Calculus of Variations Equation The fundamental partial differential equation of the calculus of variations that a function must satisfy to make a functional stationary. (d / dt) (∂L / ∂q̇_i) - (∂L / ∂q_i) = 0

Fast Fourier Transform (FFT) Mathematical & Computational Physics Computational FFT Algorithm An efficient algorithm that computes the Discrete Fourier Transform (DFT) of a sequence in O(N log N) operations instead of O(N²). Cooley-Tukey Divide-and-Conquer: O(N log N) vs O(N²)

Finite Difference Method Mathematical & Computational Physics Grid Discretization Method A numerical method for solving differential equations by approximating derivatives with finite difference quotients on a discretized grid. f'(x) ≈ (f(x + h) - f(x - h)) / (2 h), f''(x) ≈ (f(x+h) - 2f(x) + f(x-h)) / h²

Finite Element Method (FEM) Mathematical & Computational Physics Numerical Mesh Method A numerical technique for solving complex partial differential equations by subdividing a large physical domain into smaller, simpler finite sub-domains called elements. K u = f (Global Stiffness Matrix Equation)

Fourier Series Mathematical & Computational Physics Harmonic Series Expansion An expansion of a periodic function f(x) into an infinite sum of orthogonal sine and cosine harmonic functions. f(x) = a₀/2 + ∑_(n=1)^(∞) [a_n cos(n x) + b_n sin(n x)]

Fourier Transform Mathematical & Computational Physics Inverse Domain Unit (e.g. s⁻¹ or m⁻¹) An integral transform that decomposes a continuous time or spatial signal into its constituent frequency or wavevector spectrum. f̂(k) = ∫_(-∞)^(∞) f(x) e^(-i k x) dx, f(x) = (1 / 2π) ∫_(-∞)^(∞) f̂(k) e^(i k x) dk

Genetic Algorithms Mathematical & Computational Physics Stochastic Heuristic Optimization A search heuristic optimization method inspired by natural biological evolution that iteratively evolves candidate solutions toward optimal physics configurations. Evolutionary Operators: Selection, Crossover, Mutation