Mathematical & Computational Physics — 5 Terms Defined

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