Problem 1
It makes exact forms closed and creates the cochain complex whose cohomology is de Rham cohomology.
Faculty solution guide
The Exterior Derivative
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 Exterior Derivative.
Problem 4
The argument must address proving d²=0 and computing coordinate formulas 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.