$\;{u|}_{\partial \Omega} =0\;$ implies $\; \frac{\partial u}{\partial τ}=0\;$

Question

I'm reading the following Theorem of boundary estimate for the gradient:

If $\;\begin{cases} -\sum_{i,j}a_{ij}u_{x_i x_j}+\sum_{i}b_iu_{x_i}+cu=f\;,\;x\in \Omega \subset \mathbb R^n\\ u=0\;,\;x\in\partial \Omega\; \end{cases}\;$

where $\;a_{ij},b_i,c,f\;$ are bounded functions,$\;\partial \Omega\;$ satisfies in each point the condition of exterior ball, $\;c\ge 0\;$ and the operator is elliptic.

Prove there exists constant $\;C\;$ such that : $\;{\sup}_{\partial \Omega} \vert \nabla u \vert \le C\;$

In the first line of the proof, there is the claim in the title of my question. To be more specific, it says:

Since $\;u=0\;$ along $\;\partial \Omega\;$ then we only care for the derivative in the vertical direction hence $\;\vert \nabla u \vert =\vert \frac{\partial u}{\partial v} \vert\;$

I don't understand:

1. Why $\;\nabla u =(\frac{\partial u}{\partial τ},\frac{\partial u}{\partial v})\;$? Maybe a graph, would be a great help here...
2. Why $\;{u|}_{\partial \Omega} =0\;$ implies $\; \frac{\partial u}{\partial τ}=0\;$

I've been really stuck here so any help would be valuable.

Thanks in advance!