Prove that $h(p,u) = \nabla_p e(p,u)$ is implied by Roy's identity
a) The convention, when differentiating a real valued function $v$ wrt to a column vector $p$ of dimension $(G \times 1)$, is that:
$$
\nabla _{p}v\left( p,w\right) \equiv \frac{\partial v}{\partial p}\left(
p,w\right) =\left( 
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,w\right)  \\ 
\vdots  \\ 
\frac{\partial v}{\partial p_{G}}\left( p,w\right) 
\end{array}%
\right). 
$$

b) When expenditures $w$ are not constant, the column vector $p$ with respect to which we take the derivative occurs twice in $v(p, e(p,\overline{u}))$, and the chain rule has to be applied in order to obtain the total effect of the price change on the indirect utility: the first partial effect on utility is due to the price changes, and the second effect is through the adjustment in expenditures implied by the price change. Hence the expression:
\begin{eqnarray*}
&amp;&amp;\frac{\partial v}{\partial p}\left( p,e\left( p,u\right) \right) +\frac{%
\partial v}{\partial w}\left( p,e\left( p,u\right) \right) \frac{\partial e}{%
\partial p}\left( p,u\right)  \\
&amp;=&amp;\left( 
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,e(p,u)\right)  \\ 
\vdots  \\ 
\frac{\partial v}{\partial p_{G}}\left( p,e(p,u)\right) 
\end{array}%
\right) +\frac{\partial v}{\partial w}\left( p,e(p,u)\right) \left( 
\begin{array}{c}
\frac{\partial e}{\partial p_{1}}\left( p,u\right)  \\ 
\vdots  \\ 
\frac{\partial e}{\partial p_{G}}\left( p,u\right) 
\end{array}%
\right). 
\end{eqnarray*}