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Formula for the Curvative of a Curve

```Date: 10/10/2002 at 21:10:15
Subject: Dimensions

I have an assignment in English class to convince someone who believes
a wall has only one side that there is another side to it. I was
wondering if there is a formula or theorem I can use for this?
```

```
Date: 11/17/2002 at 20:26:48
From: Doctor Nitrogen
Subject: Re: Dimensions

Suppose both of you are in a very large room with a very, very long
wall.

1. Imagine both of you traveling along the long length of the wall. If
the wall is perfectly flat, you might never be able to convince the
other party there is another side to the wall.

2. If the very long wall is curved away from both of you as you travel
alongside it inside the room, you could convince the party there must
be another side to the wall, since it is curved and curving in some
direction away from both of you and in some direction normal
(perpendicular) to the wall.

3. If the very long wall is curved inward toward the room and toward
both of you, you could convince the other party there must be another
side to the wall, since it is curved and curving in some direction
toward both of you and in some direction normal (perpendicular) to the
wall.

If you take a magic marker and draw a long curve on the curved wall
as both of you travel around the big room, you can calculate the
curvature would either point out from the big room (the wall curving
away from you both) or into the big room (the wall curving inward
toward you both).

The direction the radius of curvature would be along would be
perpendicular to the direction in which you both are traveling, and
perpendicular to the up or down direction of the room, so for a
curved wall you could hypothesize there lies another side to the
wall.

In Calculus the following formula computes such curvature for a
curve drawn on a curved surface:

Curvature  K = y"/[1 + (y')^2]3/2.

y' is the derivative of y with respect to x, and y" is the second
derivative of y with respect to x.

Forum/Doctor Math whenever you have any math-related questions.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
College Higher-Dimensional Geometry
High School Calculus
High School Higher-Dimensional Geometry

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