I'm trying to understand vector notation, in particular when operators act on one another. I'm assuming the parenthesis are important to preserve the physical meaning of the operator. How do you know when to keep/introduce parenthesis? Can some one help expand the following expression?
$(\textbf{A} \cdot \nabla)(\textbf{A} \cdot \nabla)$
NOTE: $\textbf{A}$ is a vector in $\text{R}^3$ and I'm dealing with basic Euclidean space, where $\nabla$ is defined as the gradient in a conventional sense.
In vector calculus it is common to interpret the differential operator $\nabla$ as if it were a vector itself, i.e. $ \nabla = (\partial_1, \partial_2, \partial_3)^T$. From this point of view it presents no problem to take the scalar product $A \cdot \nabla$, as $\nabla$ is just a vector: $$ A \cdot \nabla = A_1 \partial_1 + A_2\partial_2 + A_3\partial_3. $$ Note that this is giving you a new differential operator, which you could apply to either a scalar function $\alpha(x_1, x_2, x_3)$, i.e. $$ (A \cdot \nabla)\alpha = A_1 (\partial_1 \alpha) + A_2 (\partial_2 \alpha) + A_3(\partial_3 \alpha) \tag{$\ast$} $$ or componentwise to a vector.
Interpreting the differential operator $A \cdot \nabla$ as scalar, we can insert it into ($\ast$) and get $$ (A \cdot \nabla)\,(A \cdot \nabla) = A_1\partial_1 \big(A_1 \partial_1 + A_2 \partial_2 + A_3\partial_3 \big) + \dots = A_1^2 \partial_1^2 + A_1 A_2 \partial_1\partial_2 + A_1 A_3 \partial_1\partial_3 \cdots $$ If $A$ is a vector field you will have to take into account derivatives of $A$ as well (product rule).