Problem 1
If q is a regular value of F:M→N, then F⁻¹(q) is an embedded submanifold of codimension dim N.
Faculty solution guide
Regular Values and the Preimage Theorem
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 Values and the Preimage Theorem.
Problem 4
The argument must address proving level sets are embedded submanifolds with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in constraint manifolds, bifurcations, intersections, robotics, optimization, and dynamical systems, 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.