# Curvature

• Apr 16th 2012, 09:11 PM
kayla12345
Curvature
r(t) = < e^t * cost, e^t*sint, 5>
Find the curvature at the point (x, y, z) = (-e^π, 0, 5)
explain briefly how you come up with answer and also express answer as a decimal.
• Apr 17th 2012, 05:24 AM
MathoMan
Re: Curvature
If a point $T(x,y,z)$ belongs to a curve $\vec{r}(t)=x(t)\vec{i}+y(t)\vec{j}+z(t)\vec{k}$ then exists a parameter value $t\in \mathbb{R}$ such that
$x=x(t), \, y=y(t), \, z=z(t)$. In case of a given point $t=-\pi.$

Wikipedia says that for a space curve its curvature is given by

All derivatives are with respect to variable t. Here $\kappa$ is a function in variable $t$, $\kappa=\kappa (t).$ Find the expression for $\kappa(t)$ and evaluate it for $t=-\pi$.
• Apr 17th 2012, 05:55 AM
tom@ballooncalculus
Re: Curvature
Also, notice that as z is constant you are really looking at,

http://www.ballooncalculus.org/draw/graph/curvature.png

... so you only need the 2-D version (Curvature - Wikipedia, the free encyclopedia).

Just in case a picture helps differentiate...

http://www.ballooncalculus.org/draw/diffProd/eight.png

... where (key in spoiler) ...

Spoiler:
http://www.ballooncalculus.org/asy/prod.png

... is the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to t. And,

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