Problem 1
Their velocity field is parallel along the curve: ∇_{γ′}γ′=0.
Faculty solution guide
Geodesics and the Geodesic Equation
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Geodesics and the Geodesic Equation.
Problem 4
The argument must address deriving locally length-stationary curves with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in general relativity, navigation, shape spaces, mechanics, optimization, and machine learning on manifolds, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.