Problem 1
It is the linear map between tangent spaces that gives the first-order behavior of F at p.
Faculty solution guide
Differentials, Rank and the Chain Rule
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 Differentials, Rank and the Chain Rule.
Problem 4
The argument must address computing pushforwards and interpreting rank geometrically with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in mechanics, relativity, robotics, optimization on manifolds, and geometric data analysis, 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.