Problem 1
At every intersection point p, T_pA+T_pB=T_pM.
Faculty solution guide
Transversality and Stable Intersections
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 Transversality and Stable Intersections.
Problem 4
The argument must address computing tangent-space sums and expected dimensions 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.