I was given the following question which I am unable to get a seemingly correct answer from:
A body expands linearly by a factor $\alpha$ due to an increase in temperature. Because of the expansion, a point at $\vec{r}$ is displaced to the point $\vec{r}+\vec{h}$. Calculate $\nabla \cdot \vec{h}$. By what fraction does the volume increase?
From the question we can deduce that:
$$\vec{r}+\vec{h}=\alpha\vec{r}$$
We can take the divergence of both sides and use the linearity of $\nabla$ to write:
$$\nabla \cdot (\vec{r}+\vec{h})=\alpha \nabla \cdot \vec{r}\implies \nabla \cdot \vec{r} + \nabla \cdot \vec{h} = \alpha(\nabla \cdot \vec{r})$$
We can then rearrange and use $\nabla \cdot \vec{r}=3$ to get:
$$\nabla \cdot \vec{h}=3(\alpha - 1)$$
However, I am unsure how to use this to calculate the fractional volume increase. I tried:
$$\frac{V_{\text{new}}}{V_{\text{old}}}=\frac{\iiint_{V}\nabla\cdot\vec{h}\:\mathrm{d}V}{\iiint_{V}\:\mathrm{d}V}=3(\alpha - 1)$$
However, this is clearly wrong as we would expect the answer to be $\frac{V_{\text{new}}}{V_{\text{old}}} \in \mathcal{O}(\alpha^{3})$, so I'd appreciate any hints to get me back on the right track.
Thanks in advance!