Help on dot and wedge symbols

1) You know that $\nabla$ is a differential operator, right? Which means that $\nabla . u \neq u.\nabla$, just as $\frac{d}{dx}u \neq u \frac{d}{dx}$.

2) In this context, $A \wedge B$ is almost certainly the same as $A \times B$; it's just the vector cross product. (In 3-space, this is basically also the same as the exterior wedge product, but you probably don't want to think of it like that).

1) I know that the first is just the divergence, and I'm aware of the faulty notation since it 'views' as a dot product but no true product is taking place. Rather, the operators are acting on each component of the vector, respectively. I'm still lost on the meaning of $u\cdot\nabla$

I'm working in 3D, so could it be that I multiply the matrices as normal but the dot is to let me know that I allow the operators to act on the entries rather than true multiplication. Meaning $\nabla\cdot u$ would give me a $1\times1$ matrix but $u\cdot\nabla$ would give me a $3\times3$ matrix? That's all I've been able to come up with so far.

2) I googled 'cross product' with the suggestion, and I didn't realize the wedge is used in physics notation. I'm good on this one now. Thanks.

The operation of taking two vectors and forming a matrix is called the outer product (not to be confused with the exterior product), usually denoted $\otimes$; I really don't think it would be appropriate to denote it by the same symbol as the inner product. Now I'm just guessing, but I think $(u \cdot \nabla) f$ is $\displaystyle u_x \frac{\partial f}{\partial x} + u_y \frac{\partial f}{\partial y} + u_z \frac{\partial f}{\partial z}$.

Also note that $\nabla$ operates on a vector. but $(u \cdot \nabla)$ operates on a scalar.

I'm almost completely sold on treating it as the normal dot product if $u \cdot \nabla$ can act on a vector. Wouldn't it be ok to just apply $u \cdot \nabla$ to each component of a vector or would that present a problem?