How to interpret advection (v.del) v->v.(del v)

[Monty Hall]

I'm looking @ convective accerlation term in http://en.wikipedia.org/wiki/Navier_stokes_equation#Convective_acceleration. I don't understand the terminology. If v is a vector, it says that $$(\mathbf{v}\cdot\nabla)\mathbf{v}$$ can be written as $$\mathbf{v}\cdot\nabla \mathbf{v}$$. I thought that $$\nabla \mathbf{v}$$ is the transpose of the Jacobian matrix for $$\mathbf{v}$$. As I'm not familiar with the terminology it almost looks like ($$\mathbf{v} \cdot \nabla \mathbf{v}$$ = vector . matrix), which can't be right. However, it appears $$(v\cdot\nabla)v$$ is a vector. Can somebody shed some light on how $$v\cdot\nabla v$$ is a vector? If $$(\mathbf{v}\cdot\nabla)\mathbf{v}=\mathbf{v}\cdot\nabla \mathbf{v}$$ & $$\nabla \mathbf{v} = (\mathbf{J}\mathbf{v})^T$$. I'm sure I'm mutilating the terminology, if anybody could shed light on this, much appreciated.

 

[arildno]

In index notation, we have:
$$\vec{v}\cdot\nabla=\sum_{j=1}^{3}v_{j}\frac{\partial}{\partial{x}_{j}}$$
where the indices ought to be fairly self-evident.
Usually, we just omit the summation symbol, writing $$v_{j}\frac{\partial}{\partial{x}_{j}}$$ instead.

Setting this alongside a vector $v_{i}$ then, we have:
$$(\vec{v}\cdot\nabla)\vec{v}=(v_{j}\frac{\partial}{\partial{x}_{j}})v_{i}=v_{j}\frac{\partial{v}_{i}}{\partial{x}_{j}}$$

The quantity $$\frac{\partial{v}_{i}}{\partial{x}_{j}}$$ is a matrix

 

[Monty Hall]

Thanks for your reply, I've seen that as well the (v.del) makes a lot more sense. I know (v.del)v should equal v.del(v), but I'm curious how can I come to same result starting with v.del(v) which to me looks like a vector.matrix.