Problem 1
A regular orientation-preserving reparametrization changes speed and parameter measure inversely, leaving the integral of speed unchanged.
Faculty solution guide
Regular Curves and Arc-Length Parametrization
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 Regular Curves and Arc-Length Parametrization.
Problem 4
The argument must address reparametrizing curves by distance and proving invariance of length with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in computer graphics, architecture, materials, trajectories, surface reconstruction, and shape optimization, 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.