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. 40

The converse problem is somewhat more complicated: Find the curves which admit a given curve C as involute. Such curves are called evolutes of C (German: Evolute; French: développées). Their tangents are normal to C({\bf x}) and we can therefore write the equation of the evolute {\bf y} (Fig. 1-34):
{\bf y} = {\bf x} + a_1{\bf n} + a_2{\bf b}.
Hence
{d{\bf y}\over ds} = {\bf t}(1-a_1\kappa) + {\bf n}\left({da_1\over ds}-\tau a_2\right) + {\bf b}\left({da_2\over ds}+\tau a_1\right)
must have the direction of a_1{\bf n} + a_2{\bf b}, this tangent to the evolute:
\kappa = 1/a, \qquad R= a_1,
and
{{da_1\over ds} - \tau a_2\over a_1} = {{da_2\over ds}+\tau a_1\over a_2},

which can be written in the form:

{a_2{dR\over ds} - R{da_2\over ds} \over a_2^2 + R^2} = \tau.

This expression can be integrated:
\tan^{-1}{R\over a_2} = \int \tau\,ds + {\rm const},
or
a_2 = R\left[{\rm cot}\left(\int \tau\,ds + {\rm const}\right)\right].

The equation of the evolute is:

{\bf y} = {\bf x} + R\left[{\bf n} + {\rm cot}\left(\int \tau\,ds + {\rm const}\right){\bf b}\right].

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.