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.