Pen Settings



CSS Base

Vendor Prefixing

Add External Stylesheets/Pens

Any URL's added here will be added as <link>s in order, and before the CSS in the editor. If you link to another Pen, it will include the CSS from that Pen. If the preprocessor matches, it will attempt to combine them before processing.

+ add another resource


Babel is required to process package imports. If you need a different preprocessor remove all packages first.

Add External Scripts/Pens

Any URL's added here will be added as <script>s in order, and run before the JavaScript in the editor. You can use the URL of any other Pen and it will include the JavaScript from that Pen.

+ add another resource


Save Automatically?

If active, Pens will autosave every 30 seconds after being saved once.

Auto-Updating Preview

If enabled, the preview panel updates automatically as you code. If disabled, use the "Run" button to update.

Format on Save

If enabled, your code will be formatted when you actively save your Pen. Note: your code becomes un-folded during formatting.

Editor Settings

Code Indentation

Want to change your Syntax Highlighting theme, Fonts and more?

Visit your global Editor Settings.


<script type="text/x-mathjax-config">
MathJax.Ajax.config.path["Contrib"] = ""; 
  tex2jax: {inlineMath: [['$','$'],['\\(','\\)']]},
  extensions: ["[Contrib]/a11y/accessibility-menu.js"],
  menuSettings: {
    explorer: true
<script src=""></script>

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

<h2>p. 23</h2>
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.
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 +

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}
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.