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\centerline
{
\bf Dr. Z's Math251 Handout \#13.4 (2nd ed.) [Curvature]
}

By Doron Zeilberger


{\bf Problem Type 13.4a}: Find the curvature for 
$$
{\bf r}(t)=x(t) \, {\bf i}+ y(t) \, {\bf j}+z(t) \, {\bf k} \quad.
$$


{\bf Example Problem 13.4a}: Find the curvature for 
$$
{\bf r}(t)=t \, {\bf i}+ 2t \, {\bf j}+t^2 \, {\bf k} \quad.
$$
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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} Compute ${\bf r}'(t)$ and ${\bf r}''(t)$ .
&
{\bf 1.} 
$$
{\bf r}'(t)={\bf i}+ 2 \, {\bf j}+2t \, {\bf k} \quad.
$$
$$
{\bf r}''(t)=2 \, {\bf k} \quad.
$$


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{\bf 2.} Compute the {\bf cross product}

${\bf r}'(t) \times {\bf r}''(t)$.
&
{\bf 2.} ${\bf r}'(t) \times {\bf r}''(t)$ equals
$$
\left \vert \matrix{ {\bf i} & {\bf j} & {\bf k}  \cr
1 & 2 & 2t\cr
0 & 0 & 2 
} \right \vert \quad =
$$
$$
{\bf i} \left \vert \matrix{ 
2 & 2t\cr
0 & 2 
} \right \vert 
-{\bf j} \left \vert \matrix{ 
1 & 2t\cr
0 & 2 
} \right \vert 
+{\bf k} \left \vert \matrix{ 
1 & 2 \cr
0 & 0 
} \right \vert 
$$
$$
=4 \, {\bf i}-2 \,{\bf j}
$$
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{\bf 3.} Find the magnitude of the vector that you found in
step 2 (namely ${\bf r}'(t) \times {\bf r}''(t)$).
Also find the magnitude of ${\bf r}'(t)$, and
finally use the formula for the curvature
$$
\kappa (t)=
{{\vert {\bf r}'(t) \times {\bf r}''(t) \vert} \over
{\vert {\bf r}'(t) \vert^3}}
$$
&
{\bf 3.} 
$$
\vert {\bf r}'(t) \times {\bf r}''(t) \vert=
\sqrt{4^2+2^2+0^2}=\sqrt{20} \quad.
$$
$$
\vert {\bf r}'(t)\vert=
\vert {\bf i}+ 2 \, {\bf j}+2t \, {\bf k}\vert 
$$
$$
=\sqrt{1^2+2^2+(2t)^2}=\sqrt{5+4t^2} \quad .
$$
Finally,
$$
\kappa (t)=
{{\sqrt{20}} \over
{(\sqrt{5+4t^2})^3 } }
$$
This is the {\bf Ans.}.
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{\bf Problem from a previous Final Exam}

Find the curvature of the curve
$$
{\bf r} (t)= \langle t, t^2, {{2} \over {3}}t^3 \rangle 
$$
at the point $(1,1,{{2} \over {3}})$.

{\bf Ans.}: ${{2} \over {9}}$.

\end


