 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2014/4/c/solution

C^{1}

f : R^{n} \to R^{m}

with surjective derivative at every point of

f^{- 1} (0)

has a zero set that is an embedded

C^{1}

n - m

. At each point

a

of that zero set,

T_{a} (f^{- 1} (0)) = ker D f_{a} .

For a smooth map the submanifold is smooth. The full-rank hypothesis is essential for this kernel description when using a particular defining map.

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