Problem Solving Guide

Differential Geometry · Axiom Academy

SUMMARY Differential Geometry Problem Solving Guide A comprehensive reference for tackling common problem types with strategic approaches and decision frameworks. Computing Curvature: Find κ = |T'(s)| where s is arc length, or use κ = |α' × α''|/|α'|³ for arbitrary parametrization Computing Torsion: Calculate τ = (α' × α'') · α'''/|α' × α''|² or use the Frenet-Serret formulas Frenet Frame: Normalize T = α'/|α'| , find N = T'/|T'| , then B = T × N Reconstruction: Given κ(s) and τ(s) , solve Frenet-Serret equations as a system of ODEs Finding Parametrizations: Identify natural coordinates (spherical, cylindrical) or use level set equations to solve for one variable First Fundamental Form: Compute E = X u ·X u , F = X u ·X v , G = X v ·X v Second Fundamental Form: Find normal N = X u × X v /|X u × X v | , then L = X uu ·N, M = X uv ·N, N = X vv ·N Curvatures: Gaussian K = (LN - M²)/(EG - F²) , Mean H = (EN - 2FM + GL)/(2(EG - F²)) Decision Tree: Choosing Curve Approach Setting Up Equations: Use Euler-Lagrange equations on E = ∫√(Eu'² + 2Fu'v' + Gv'²)ds Christoffel Symbols Method: Solve u'' + Γ 1 ij u'u' = 0 and similar for v Exploiting Symmetry: If surface has rotational symmetry, use Clairaut's theorem: r sin θ = constant Special Cases: Great circles on spheres, straight lines on planes, helices on cylinders Christoffel Symbols: Use Γ k ij = ½g kl (∂ i g jl + ∂ j g il - ∂ l g ij ) Matrix Inversion: Compute g ij from first fundamental form matrix inverse

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