The Prelims Formula Shiiiiiiit

Orbital acceleration in polar: a = (r̈ − rθ̇2)r̂ + (rθ̈ + 2ṙθ̇)θ̂ Legendre polynomials: P1(x) = 1, P2(x) = x, P3(x) = (3x 2 − 1)/2; orthogonality: ∫ 1 −1 Pm(x)Pn(x)dx = 2 2n+1δmn Eccentricity: e > 1 hyperbola; e = 1 parabola; 0 < e < 1 ellipse; e = 0 circle Stirling’s Approximation: lnN ! = N lnN −N Gaussian Integral: ∫∞ −∞ e −ax2dx = √ π/a ∫∞ −∞ x 2e−ax 2 dx = √ π/2a3/2 ∫∞ 0 x 3e−αx 2 = 1/2α2 Delta function idenitity: ∫ f(x)δ′(x− a)dx = − ∫ f ′(x)δ(x− a) = −f ′(a) Finite geometric sum: ∑n−1 k=0 ar k = a(1−r n) 1−r Memorize the metric ds2 = h1dx 2 1 + h 2 2dx 2 2 + h 2 3dx 2 3 for spherical, cylinder coordinates. Gradient, divergence, curl, and Laplacian are derived below: ∇φ = 1 h1 ∂φ ∂x1 ê1 + 1 h2 ∂φ ∂x2 ê2 + 1 h3 ∂φ ∂x3 ê3 ∇ ·A = 1 h1h2h3 ( ∂ ∂x1 (h2h3A1) + ∂ ∂x1 (h1h3A2) + ∂ ∂x1 (h1h2A3) )