Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 26G / a / Solution

A point $y_0\in Y$ is a regular value of the smooth map $f:X\to Y$ if, for every $x\in f^{-1}(y_0)$, the derivativeis surjective. A value outside the image is regular vacuously.

For a nonvacuous example, takeThe value $1$ is regular: its fibre is the unit circle, and at every $(x,y)$ on that circle,is a nonzero linear functional and hence surjective.

Sard theorem states that the set of critical values of a smooth map between smooth manifolds has measure zero in the target. Equivalently, almost every target point is a regular value.

Solved by gpt-5.6-sol high.