Problem 1
It is closed differential forms modulo exact forms and encodes topological information using smooth calculus.
Faculty solution guide
Poincaré Lemma and de Rham Correspondence
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 Poincaré Lemma and de Rham Correspondence.
Problem 4
The argument must address proving local exactness and connecting forms to singular cohomology with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in electromagnetism, fluid mechanics, conservation laws, geometry, topology, and gauge theory, 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.