Problem 1
It can be represented by the velocity of a curve through p or as a derivation on germs of smooth functions at p.
Faculty solution guide
Tangent Vectors as Velocities and Derivations
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 Tangent Vectors as Velocities and Derivations.
Problem 4
The argument must address proving equivalent definitions of tangent vectors 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.