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<h1>An example from Struik's <i>Lectures on Classical Differential Geometry</i></h1>

<h2>p. 23</h2>
<p>
This procedure is simply a generalization of the method used in Sects.
1-3 and 1-4 to obtain the equations of the osculating plane and the
osculating circle.  Let $f(u)$ near $P(u=u_0)$ have finite derivatives
$f^{(i)}(u_0)$, $i = 1, 2, \ldots, n+1$.  Then if we take $u=u_1$ at $A$
and write $h = u_1 - u_0$, then there exists a Taylor development of $f(u)$
of the form (compare Eq.  (1-5)):
$$f(u_1) = f(u_0) + hf'(u_0)+{h^2\over 2!}f''(u_0) + \cdots + {h^{n+1}\over (n+1)!}f^{(n+1)}(u_0) + o(h^{n+1}).$$

Here, $f(u_0)=0$ since $P$ lies on $\Sigma_2$, and $h$ is of order $AP$
(see theorem Sec.  1-2); $f(u_1)$ is of order $AD$.  <i>Hence necessary and
sufficient conditions that the surface has a contact of order $n$ at $P$
with the curve are that at $P$ the relations hold:</i>

$$f(u) = f'(u) = f''(u) = \cdots = f^{(n)}(u) = 0;\quad f^{(n+1)}(u) \ne 0.$$

<hr>
<h2>
p. 154
</h2>

If $P(u,v)$ and $Q(u,v)$ are two functions of $u$ and $v$ on a surface,
then according to Green's theorem and the expression in Chapter 2, Eq.
(3-4) for the element area:

$$\int_C P\,du + Q\, dv = \int\!\!\!\int_A \left({\partial Q\over \partial u} - {\partial P\over \partial v}\right) {1\over \sqrt{EG-F^2}}\,dA,$$
where $dA$ is the element of area of the region $R$ enclosed by the curve
$C$.  With the aid of this theorem we shall evaluate

$$\int_C \kappa_g\,ds,$$

where $\kappa_g$ is the geodesic curvature of the curve $C$.  If $C$ at a
point $P$ makes the angle $\theta$ with the coordinate curve $v = {\rm constant}$ and if the coordinate curves are orthogonal, then, according to
Liouville's formula (1-13):
$$\kappa_g\,ds = d\theta + \kappa_1(\cos\theta)\,ds + \kappa_2(\sin\theta)\,ds.$$

Here, $\kappa_1$ and $\kappa_2$ are the geodesic curvatures of the curves
$v = {\rm constant}$ and $u = {\rm constant}$ respectively.  Since
$$\cos\theta\,ds = \sqrt{E}\,du, \qquad \sin\theta\,ds = \sqrt{G}\,dv,$$
we find by application of Green's theorem:
$$\int_C\kappa_g\,ds = \int_C d\theta + \int\!\!\!\int_A\left({\partial\over\partial u} \left(\kappa_2\sqrt{G}\,\right) - {\partial\over \partial v}\left(\kappa_1\sqrt{E}\,\right)\right)\,du\,dv.$$

<p>
The Gaussian curvature can be written, according to Chapter 3, Eq. (3-7),
$$K = -{1\over 2\sqrt{EG}} \left[{\partial\over\partial u}{G_u\over \sqrt{EG}} + {\partial\over\partial v}{E_v\over\sqrt{EG}}\right] ={1\over\sqrt{EG}}\left[ -{\partial\over\partial u} \left(\kappa_2\sqrt{G}\,\right) + {\partial\over\partial v} \left(\kappa_1\sqrt{E}\,\right)\right],$$
so we obtain the formula
$$\int_C\kappa_g\,ds = \int_C d\theta - \int\!\!\!\int_A K\,dA.$$

The integral $\int\!\!\int_A K\,dA$ is known as the <i>total</i> or
<i>integral curvature</i>, or <i>curvature integra</i>, of the region $R$,
the name by which Gauss introduced it.



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