Problem 1
Gaussian curvature can be computed from the first fundamental form, so it is intrinsic despite being defined through an embedding.
Faculty solution guide
Gaussian Curvature, Mean Curvature and Theorema Egregium
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 Gaussian Curvature, Mean Curvature and Theorema Egregium.
Problem 4
The argument must address comparing intrinsic and extrinsic bending 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.