Mathematical & Computational Physics

Algorithm

A finite, unambiguous sequence of mathematical instructions implemented in software to solve specific physical equations or compute…

Step-by-step computational procedure Computational Logic

Bessel 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) Cylindrical Special Functions

Mathematical & Computational Physics

Big O Notation

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 Algorithmic Asymptotic Complexity

Mathematical & Computational Physics

Boundary Value Problem

A differential equation together with a set of additional constraints, called boundary conditions, specified at the boundaries of the…

Dirichlet: y(a)=y_A; Neumann: y'(a)=y'_A; Robin: α y(a) + β y'(a) = γ Differential Equation Condition

Mathematical & Computational Physics

Canonical 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 Phase Space Coordinate Transformation

Mathematical & Computational Physics

Chaos Theory

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 Divergence Rate λ (s⁻¹)

Mathematical & Computational Physics

Complex Analysis

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) Complex Function Theory

Mathematical & Computational Physics

Curl

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]| [Vector Field Unit] / m

Mathematical & Computational Physics

Curve Fitting

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 Goodness of Fit χ²

Mathematical & Computational Physics

Differential Equation

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 Mathematical Equation Class

Mathematical & Computational Physics

Dirac Delta Function

A generalized function (distribution) that is zero everywhere except at x = 0 where it is infinite, with an integral over all space equal…

δ(x) = 0 for x ≠ 0, ∫_(-∞)^(∞) δ(x) dx = 1, ∫ f(x) δ(x - a) dx = f(a) 1 / [x Unit]

Mathematical & Computational Physics

Divergence

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) [Vector Field Unit] / m

Mathematical & Computational Physics

Divergence Theorem

A fundamental theorem of vector calculus stating that the outward flux of a vector field through a closed surface ∂V equals the volume…

∯_(∂V) F · dA = ∭_V (∇ · F) dV Vector Integral Theorem

Mathematical & Computational Physics

Eigenvalue Problem

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…

Â v = λ v, det(A - λ I) = 0 Eigenvalues λ (Scalar Units)

Mathematical & Computational Physics

Eigenvalues and Eigenvectors

Special vectors v associated with a linear transformation matrix A that maintain their spatial direction when operated on, scaled only by…

A v = λ v (where v ≠ 0) Linear Algebra Object

Mathematical & Computational Physics

Error Propagation

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) Uncertainty Units

Mathematical & Computational Physics

Euler Method

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) First-Order Numerical ODE Integration

Mathematical & Computational Physics

Euler-Lagrange 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 Calculus of Variations Equation

Mathematical & Computational Physics

Fast Fourier Transform (FFT)

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) Computational FFT Algorithm

Mathematical & Computational Physics

Finite Difference 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 Grid Discretization Method

Mathematical & Computational Physics

Finite Element Method (FEM)

A numerical technique for solving complex partial differential equations by subdividing a large physical domain into smaller, simpler…

K u = f (Global Stiffness Matrix Equation) Numerical Mesh Method

Mathematical & Computational Physics

Fourier Series

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)] Harmonic Series Expansion

Mathematical & Computational Physics

Fourier Transform

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 Inverse Domain Unit (e.g. s⁻¹ or m⁻¹)

Mathematical & Computational Physics

Genetic Algorithms

A search heuristic optimization method inspired by natural biological evolution that iteratively evolves candidate solutions toward optimal…

Evolutionary Operators: Selection, Crossover, Mutation Stochastic Heuristic Optimization