 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/26f/c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

F (x, y, z) = x^{2} + y^{2} + z^{2} - 2 x y - 2 yz .

\nabla F = 2 (x - y, y - x - z, z - y) .

If this vanishes, the first and third coordinates give

x = y = z

, and the middle coordinate then gives

y = 0

. Thus the origin is the only critical point of

F

. It has been removed from

V

, so

0

is a regular value of the restriction under consideration. The regular level set theorem shows that

V = F^{- 1} (0) ∖ {0}

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