What is the expanded form of (v ∙ grad)w?

Since we have a dot product involving v, I assume that v is a vector but then (v- grad) evaluates to a scalar operator and w could be either a vector or a scalar.

Let \(\displaystyle v= v_x\vec{i}+ v_y\vec{j}+ v_z\vec{k}\) and \(\displaystyle v\cdot\nabla= v_x\frac{\partial}{\partial x}+ v_y\frac{\partial}{\partial y}+ v_z\frac{\partial}{\partial z}\)

If w is a vector, \(\displaystyle w= w_x\vec{i}+ w_y\vec{j}+ w_z\vec{k}\) then this is a scalar product- the scalar multiplying each part of the vector:

\(\displaystyle (v_x\frac{\partial w_x}{\partial x}+ v_y\frac{\partial w_x}{\partial y}+ v_z\frac{\partial w_x}{\partial z})\vec{i}\)\(\displaystyle + ( v_x\frac{\partial w_y}{\partial x}+ v_y\frac{\partial w_y}{\partial y}+ v_z\frac{\partial w_y}{\partial z})\vec{j}\)\(\displaystyle + ( v_x\frac{\partial w_z}{\partial x}+ v_y\frac{\partial w_z}{\partial y}+ v_z\frac{\partial w_z}{\partial z})\vec{k}\)

If w is a scalar function then this is a product of two scalars:

\(\displaystyle v_x\frac{\partial w}{\partial x}+ v_y\frac{\partial w}{\partial y}+ v_z\frac{\partial w}{\partial z}\)