Mathematical & Computational Physics

Gradient

A vector operator that points in the direction of the greatest rate of increase of a scalar field, with magnitude equal to that maximum…

∇f = (∂f/∂x) î + (∂f/∂y) ĵ + (∂f/∂z) k̂ [Scalar Field Unit] / m

Gradient Descent

An iterative first-order optimization algorithm for finding local minima of a differentiable scalar objective function by taking steps…

x_(k+1) = x_k - α ∇f(x_k) Iterative Optimization Method

Mathematical & Computational Physics

Green's Function

An impulse response function used to solve inhomogeneous linear differential equations subject to specified boundary conditions.

L G(x, x') = δ(x - x'), y(x) = ∫ G(x, x') f(x') dx' Impulse Response Function

Mathematical & Computational Physics

Green's Theorem

A 2D vector calculus theorem relating a line integral around a simple closed curve C to a double area integral over the enclosed plane…

∮_C (L dx + M dy) = ∬_D ((∂M/∂x) - (∂L/∂y)) dx dy 2D Vector Integral Theorem

Mathematical & Computational Physics

Group Theory

The mathematical study of groups and algebraic symmetry structures.

Group Axioms: Closure, Associativity, Identity, Inverse Symmetry Groups (SO(3), SU(2), SU(3))

Mathematical & Computational Physics

Hamiltonian Formalism

A formulation of classical mechanics describing system dynamics in 2N-dimensional phase space using position q_i and canonical momentum p_i.

H(q, p, t) = ∑ p_i q̇_i - L, q̇_i = ∂H / ∂p_i, ṗ_i = -∂H / ∂q_i Joules (Energy)

Mathematical & Computational Physics

Hermite Polynomials

A sequence of classical orthogonal polynomials satisfying Hermite's differential equation y'' - 2x y' + 2n y = 0.

H_n(x) = (-1)^n e^(x²) (d^n / dx^n) (e^(-x²)) Dimensionless Orthogonal Polynomials

Mathematical & Computational Physics

Hessian Matrix

A square matrix of second-order partial derivatives of a scalar-valued function, describing local surface curvature and local extrema…

H_ij = ∂²f / (∂x_i ∂x_j) Second Derivative Matrix

Mathematical & Computational Physics

Hilbert Space

A complete abstract vector space possessing an inner product that allows length and angle to be defined for infinite-dimensional function…

⟨f, g⟩ = ∫ f*(x) g(x) dx, ||f|| = √(⟨f, f⟩) Complete Inner Product Space

Mathematical & Computational Physics

Jacobian Matrix

A matrix of all first-order partial derivatives of a vector-valued multivariable function, representing local linear coordinate…

J_ij = ∂f_i / ∂x_j, dV' = |det J| dV Coordinate Transformation Factor

Mathematical & Computational Physics

Lagrangian Formalism

A formulation of classical mechanics where system dynamics are derived from a single scalar function, the Lagrangian L(q, q̇, t).

L = T - V (Kinetic Energy minus Potential Energy) Joules (Action S in J·s)

Mathematical & Computational Physics

Laplace Transform

An integral transform that maps a real-valued time-domain function f(t) into a complex frequency-domain function F(s).

L{f(t)} = F(s) = ∫₀^(∞) f(t) e^(-s t) dt s = σ + i ω (Complex Frequency Domain)

Mathematical & Computational Physics

Least Squares Data Fitting

A standard mathematical regression technique for determining the best-fitting curve or line to a set of data points by minimizing the sum…

Minimizing S = ∑_(i=1)^(N) (y_i - f(x_i, β))² Statistical Regression Method

Mathematical & Computational Physics

Legendre Polynomials

A sequence of orthogonal polynomials satisfying Legendre's differential equation (1-x²) y'' - 2x y' + n(n+1) y = 0.

P_n(x) = (1 / (2^n n!)) (d^n / dx^n) ((x² - 1)^n) Dimensionless Orthogonal Polynomials

Mathematical & Computational Physics

Linear Algebra

The mathematical discipline studying vector spaces, linear transformations, matrices, and systems of linear equations.

A x = b, v · w = ∑ v_i w_i Vector Spaces & Matrices

Mathematical & Computational Physics

Machine Learning in Physics

The application of artificial intelligence, neural networks, and statistical learning models to solve complex physical equations and…

Physics-Informed Neural Networks (PINN): Loss = L_data + λ L_physics Computational AI Framework

Mathematical & Computational Physics

Matrix Mechanics

An algebraic formulation of quantum mechanics created by Werner Heisenberg in 1925 expressing physical observables as infinite-dimensional…

p̂ q̂ - q̂ p̂ = -i ℏ I, Ĥ = H(q̂, p̂) Matrix Operator Algebra

Mathematical & Computational Physics

Molecular Dynamics

A computer simulation method for analyzing the physical movements of atoms and molecules by numerically integrating Newton's equations of…

m_i d²r_i/dt² = F_i = -∇_i V(r₁, r₂, ..., r_N) Atomic Trajectory Simulation

Mathematical & Computational Physics

Monte Carlo Method

A class of computational algorithms that rely on repeated random sampling to obtain numerical solutions for complex physical systems.

I ≈ (1 / N) ∑_(i=1)^(N) f(X_i) (Central Limit Theorem) Stochastic Numerical Method

Mathematical & Computational Physics

Nonlinear Dynamics

The study of physical systems governed by non-linear equations where output changes are not directly proportional to input changes.

d x / dt = f(x) (where f is non-linear) Phase Space Dynamics

Mathematical & Computational Physics

Numerical Linear Algebra

The subfield of numerical analysis studying algorithms for performing matrix operations efficiently and accurately on digital computers.

LU: A = L U, QR: A = Q R, SVD: A = U Σ V^T Matrix Factorization Algorithms

Mathematical & Computational Physics

Numerical Stability

The property of a numerical algorithm ensuring that errors introduced during computation do not magnify uncontrollably over successive time…

Von Neumann Stability Analysis: |G(k, Δt)| ≤ 1 Algorithm Stability Property

Mathematical & Computational Physics

Parallel Computing

A computational architecture where large physical calculations are divided into smaller sub-tasks executed simultaneously across multiple…

Amdahl's Law Speedup S(P) = 1 / ((1 - p) + p / P) Speedup Factor S(P)

Mathematical & Computational Physics

Partial Differential Equations

Differential equations containing unknown multivariable functions and their partial derivatives with respect to multiple independent…

Elliptic (Laplace), Parabolic (Heat), Hyperbolic (Wave) Multivariable Differential Equations