# An example from Struik's *Lectures on Classical Differential Geometry*

## p. 23

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

##
p. 154

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:
$$
\int_C\kappa_g\,ds = \int_C d\theta +
\int\!\!\!\int_A\left({\partial\over\partial u} \left(\kappa_2\sqrt{G}\,\right) -
{\partial\over \partial v}\left(\kappa_1\sqrt{E}\,\right)\right)\,du\,dv.
$$
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 *total* or
*integral curvature*, or *curvature integra*, of the region $R$,
the name by which Gauss introduced it.