Problem 1
It determines tangent-vector lengths, angles, surface area, and intrinsic curve length.
Faculty solution guide
The First Fundamental Form
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 The First Fundamental Form.
Problem 4
The argument must address measuring length, angle, and area intrinsically 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.