Let has non-empty boundary . Let denote the unit inner normal to at . WLOG assume is given by where and and , where is a neighborhood of . Also assume is the coordinate axis.

The unit inner normal vector at a point is given by

Since , there exists a neighborhood of , say such that for , there will exist a unique point such that . Then satisfy

Prove that for each point , we have .

Since , (2) means . Then is (see GT’s book p355).

, and is given by (1)

Actually the last two terms are equal to 0. In fact.

because

Using (1)

GT’s book p355. Chapter 14. Boundary curvatures and Distance Function.

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