Answer
$$
\begin{aligned}
\tau &=\frac{2}{t^{4}+4 t^{2}+1}
\end{aligned}
$$
Work Step by Step
We find the torsion of the curve
$$
\mathrm{r}=\left\langle t, \frac{1}{2} t^{2}, \frac{1}{3} t^{3}\right\rangle
$$
We use the formula in Exercise 63(d):
$$
\tau=\frac{\left(\mathrm{r}^{\prime} \times \mathrm{r}^{\prime \prime}\right) \cdot \mathrm{r}^{\prime \prime \prime}}{\left|\mathrm{r}^{\prime} \times \mathrm{r}^{\prime \prime}\right|^{2}}
$$
We first compute the ingredients needed
$$
\mathrm{r}^{\prime}=\left\langle 1, t, t^{2}\right\rangle,\\
\mathrm{r}^{\prime \prime}=\langle 0,1,2 t\rangle,\\
\mathrm{r}^{\prime \prime \prime}=\langle 0,0,2\rangle
$$
and
$$
\begin{aligned}
\mathrm{r}^{\prime} \times \mathrm{r}^{\prime \prime}&=\left[\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & t & t^{2} \\
0, &1 & 2t\\
\end{array}\right]\\
&=\left\langle t^{2},-2 t, 1\right\rangle.\\
\end{aligned}
$$
Thus, the torsion of the given curve is
$$
\begin{aligned}
\tau &=\frac{\left(\mathrm{r}^{\prime} \times \mathrm{r}^{\prime \prime}\right) \cdot \mathrm{r}^{\prime \prime \prime}}{\left|\mathrm{r}^{\prime} \times \mathrm{r}^{\prime \prime}\right|^{2}} \\
&=\frac{\left\langle t^{2},-2 t, 1\right\rangle \cdot\langle 0,0,2\rangle}{t^{4}+4 t^{2}+1} \\
&=\frac{2}{t^{4}+4 t^{2}+1}.
\end{aligned}
$$
