I refer to such an identity as outlined here:
If $\mathbf A(\nabla \cdot \mathbf B)$ was equal to $(\mathbf A \cdot \nabla)\mathbf B$ then all terms of the first line of this identity would cancel out, leaving zero, so surely this cannot be the case? Else it would be simpler to simply write zero.
I am taught identities like this at my physics degree. According to everything I know, terms like the 1st and 4th terms should be equal. I have exhausted every attempt to find an answer to this before coming here.
Best Answer
For one, $\nabla \cdot \mathbf u \neq \mathbf u \cdot \nabla$:
The LHS is the divergence of $\mathbf u$, which is an expression, whereas the RHS is still an operator (in fact, $\mathbf u \cdot \nabla$ is called the advection operator, seen in the Navier-Stokes equations).
The issue here is that the commutative property of the dot product doesn't hold, because the dot product is supposed to be an operation between two vectors; $\nabla$ is an operator .
You're actually looking at an abuse of notation: you can interpret $\nabla \cdot \mathbf u$ intuitively, but need to be extra careful when performing algebraic manipulations.
The above also answers why the first term is not equal to the third term in your example; as for $\mathbf A(\nabla \cdot \mathbf B)$ and $(\mathbf A \cdot \nabla)\mathbf B$: the former is simply a scalar multiple of $\mathbf A$, whereas the latter is the result of some operation on the vector $\mathbf B$, which is much more complicated. Your impression that the two might be equal also involves "moving" the dot elsewhere, which can't be done either, even in the "usual" case.