Home Wikituition Browse all terms Categories
Random term

Mathematical & Computational Physics

60 terms · page 3 of 3

A–Z
All A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Filtering by: Mathematical & Computational Physics Clear all
Mathematical & Computational Physics

Perturbation Theory

A set of mathematical approximation methods for finding an approximate solution to a complex problem by starting from the exact solution of…

H = H₀ + λ H', E_n = E_n^(0) + λ E_n^(1) + λ² E_n^(2) + ... Approximation Power Series Mathematical & Computational Physics

Physical Simulation

The use of computer programs and numerical algorithms to simulate the time-dependent physical behavior of real-world or theoretical systems.

x(t + Δt) = x(t) + v(t) Δt + ½ a(t) Δt² Computational Modeling Mathematical & Computational Physics

Poisson Brackets

An algebraic operation between two phase space functions describing time evolution and conservation laws in Hamiltonian mechanics.

{f, g} = ∑ ((∂f / ∂q_i)(∂g / ∂p_i) - (∂f / ∂p_i)(∂g / ∂q_i)) Phase Space Operation Mathematical & Computational Physics

Round-off Error

The difference between the exact mathematical value of a function and the value computed using finite-precision floating-point arithmetic.

Machine Epsilon ε_mach ≈ 2⁻⁵³ ≈ 1.11 × 10⁻¹⁶ (IEEE 754 Double Precision) Floating-Point Precision Error Mathematical & Computational Physics

Runge-Kutta Methods

A family of iterative numerical integration algorithms used to solve ordinary differential equations with high accuracy.

RK4: y_(n+1) = y_n + (h / 6) (k₁ + 2 k₂ + 2 k₃ + k₄) Numerical ODE Integration Method Mathematical & Computational Physics

Spherical Harmonics

A set of special orthogonal functions defined on the 2D sphere surface serving as angular solutions to Laplace's equation in 3D spherical…

Y_l,m(θ, φ) = √(((2l+1)/(4π)) ((l-m)!/(l+m)!)) P_l^m(cos θ) e^(i m φ) Spherical Surface Functions Mathematical & Computational Physics

Statistics and Probability

The mathematical frameworks for analyzing random phenomena, quantifying measurement uncertainty, and modeling ensemble behavior in…

P(A|B) = P(B|A) P(A) / P(B), μ = E[X], σ² = Var(X) Probability (0 to 1) Mathematical & Computational Physics

Stokes' Theorem

A fundamental vector calculus theorem relating the surface integral of the curl of a vector field over an open surface S to the line…

∮_(∂S) F · dr = ∬_S (∇ × F) · dA Vector Line/Surface Integral Equality Mathematical & Computational Physics

Taylor Series

An infinite polynomial series representation of a smooth function expressed in terms of its higher-order derivatives evaluated at a single…

f(x) = ∑_(n=0)^(∞) (f^(n)(a) / n!) (x - a)^n Infinite Polynomial Expansion Mathematical & Computational Physics

Tensor

A multilinear geometric object describing linear relationships between algebraic vectors, tensors, and scalars independent of coordinate…

T'^ij = (∂x'^i / ∂x^k) (∂x'^j / ∂x^l) T^kl Geometric Multilinear Object Mathematical & Computational Physics

Variational Principle

A fundamental physical principle asserting that true physical trajectories minimize or render stationary a scalar functional (such as…

δ S = δ ∫_t₁^t₂ L(q, q̇, t) dt = 0 (Hamilton's Principle) Stationary Action Principle Mathematical & Computational Physics

Vector Calculus

A branch of mathematics dealing with multivariable differentiation and integration of vector fields in 3D Euclidean space.

∇f (Grad), ∇ · F (Div), ∇ × F (Curl), ∇²f (Laplacian) Differential Operators