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12

CHAPTER 1. THE PRINCIPAL CURVATURES.

whether we choose the normal vector to point out of the cylinder or into the cylinder. Of course, in the inward pointing case, the curvature has the opposite sign, $\mathit{k}\mathrm{=}\mathrm{-}\mathrm{1}\mathrm{/}\mathit{r}\mathrm{.}$

For inward pointing normals, the formula breaks down when $\mathit{h}\mathrm{>}\mathit{r}$, since we get multiple coverage of points in space by points of the form $\mathit{y}\mathrm{+}\mathit{t}\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{y}\mathrm{\right)}$ .

3. $\mathit{Y}$ is a sphere of radius $\mathit{R}$ with outward normal, so ${\mathit{Y}}_{\mathit{h}}$ is a spherical shell, and

${\mathit{V}}_{\mathrm{3}}\mathrm{\left(}{\mathit{Y}}_{\mathit{h}}\mathrm{\right)}\mathrm{}\mathrm{=}\mathrm{}\frac{\mathrm{4}}{\mathrm{3}}\mathit{\pi }\mathrm{\left[}\mathrm{\left(}\mathit{R}\mathrm{+}\mathit{h}{\mathrm{\right)}}^{\mathrm{3}}\mathrm{-}{\mathit{R}}^{\mathrm{3}}\mathrm{\right]}$

$\mathrm{=}\mathrm{}\mathit{h}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{2}}\mathrm{+}{\mathit{h}}^{\mathrm{2}}\mathrm{4}\mathit{\pi }\mathit{R}\mathrm{+}{\mathit{h}}^{\mathrm{3}}\frac{\mathrm{4}}{\mathrm{3}}\mathit{\pi }$

$\mathrm{=}\mathrm{}\mathit{h}\mathit{A}\mathrm{+}{\mathit{h}}^{\mathrm{2}}\frac{\mathrm{1}}{\mathit{R}}\mathit{A}\mathrm{+}{\mathit{h}}^{\mathrm{3}}\frac{\mathrm{1}}{\mathrm{3}{\mathit{R}}^{\mathrm{2}}}\mathit{A}$

$\mathrm{=}$ $\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\cdot }\mathit{A}\mathrm{\cdot }\mathrm{\left[}\mathrm{3}\mathit{h}\mathrm{+}\mathrm{3}\frac{\mathrm{1}}{\mathit{R}}$ . ${\mathit{h}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{{\mathit{R}}^{\mathrm{2}}}{\mathit{h}}^{\mathrm{3}}\mathrm{\right]}\mathrm{,}$

where $\mathit{A}\mathrm{=}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{2}}$ is the area of the sphere.

Once again, for inward pointing normals we must change the sign of the coefficient of ${\mathit{h}}^{\mathrm{2}}$ and the formula thus obtained is only correct for $\mathit{h}\mathrm{\le }\frac{\mathrm{1}}{\mathit{R}}\mathrm{.}$

So in general, we wish to make the assumption that $\mathit{h}$ is such that the map

$\mathit{Y}\mathrm{×}\mathrm{\left[}\mathrm{O}\mathrm{,}\mathrm{}\mathit{h}\mathrm{\right]}\mathrm{\to }{\mathrm{R}}^{\mathit{n}}\mathrm{,}\mathrm{}\mathrm{\left(}\mathit{y}\mathrm{,}\mathrm{}\mathit{t}\mathrm{\right)}\mathrm{↦}\mathit{y}\mathrm{+}\mathit{t}\mathit{v}\mathrm{\left(}\mathit{y}\mathrm{\right)}$

is injective. For $\mathit{Y}$ compact, there always exists an ${\mathit{h}}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ such that this condition holds for all $\mathit{h}\mathrm{<}{\mathit{h}}_{\mathrm{0}}$. This can be seen to be a consequence of the implicit function theorem. But so not to interrupt the discussion, we will take the injectivity of the map as an hypothesis, for the moment.

In a moment we will define the notion of the various averaged curvatures, ${\mathit{H}}_{\mathrm{1}}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ ${\mathit{H}}_{\mathit{n}\mathrm{-}\mathrm{1}}$, of a hypersurface, and find for the case of the sphere with outward pointing normal, that

${\mathit{H}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{1}}{\mathit{R}}\mathrm{,}\mathrm{}{\mathit{H}}_{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{R}}^{\mathrm{2}}}\mathrm{,}$

while for the case of the cylinder with outward pointing normal that

${\mathit{H}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\mathit{r}}\mathrm{,}\mathrm{}{\mathit{H}}_{\mathrm{2}}\mathrm{=}\mathrm{0}\mathrm{,}$

and for the case of the planar region that

${\mathit{H}}_{\mathrm{1}}\mathrm{=}{\mathit{H}}_{\mathrm{2}}\mathrm{=}\mathrm{0}\mathrm{.}$

We can thus write all three of the above the above formulas as

${\mathit{V}}_{\mathrm{3}}\mathrm{\left(}{\mathit{Y}}_{\mathit{h}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{3}}\mathit{A}\mathrm{\left[}\mathrm{3}\mathit{h}\mathrm{+}\mathrm{3}{\mathit{H}}_{\mathrm{1}}{\mathit{h}}^{\mathrm{2}}\mathrm{+}{\mathit{H}}_{\mathrm{2}}{\mathit{h}}^{\mathrm{3}}\mathrm{\right]}\mathrm{.}$