How do I format equations correctly? (Curl, etc.)

A question in advance: How do I format equations correctly?

Let's say

$$\mathbf{k}\cdot\nabla\times(a\cdot\mathbf{w}\frac{\partial\,\mathbf{v}}{\partial\,z})$$
- a is a scalar

Can I rewrite the expression such that

$$a\cdot\mathbf{k}\cdot\nabla\mathbf{w}\times(\frac{\partial\mathbf{v}}{\partial\,z})$$

In particular:
- Why is this possible? $$\nabla\times(\mathbf{w}\frac{\partial\mathbf{v}}{\partial\,z})=\nabla\mathbf{w}\times\frac{\partial\mathbf{v}}{\partial\,z}$$

Many thanks!

None of what you have written makes sense. If ##a## is a scalar, you cannot form the dot product of ##a## with a vector. If ##\mathbf w## and ##\mathbf v## are vectors, then ##\mathbf w \frac{\partial \mathbf v}{\partial z}## makes no sense. Nor does ##\nabla \mathbf w##.

You can only take ##a## outside the derivative if it is constant.

It's not really about "formatting"... but it's about using an identity to rewrite an expression.

Given expressions like yours that are unfamiliar,
it's probably a good idea to
write things out in component-form, and carry out the operations. even though it may be tedious.
Then look for patterns to re-group terms.
Otherwise, you're just shuffling symbols with little understanding.

The context of these equations may also be good to display.

It may be that some of your vector-calculus looking expressions are actually tensor equations

PeroK said:

None of what you have written makes sense. If ##a## is a scalar, you cannot form the dot product of ##a## with a vector. If ##\mathbf w## and ##\mathbf v## are vectors, then ##\mathbf w \frac{\partial \mathbf v}{\partial z}## makes no sense. Nor does ##\nabla \mathbf w##.

If [itex]\mathbf{a}[/itex] and [itex]\mathbf{b}[/itex] are vectors, then [itex]\mathbf{a}\mathbf{b}[/itex] is standard notation for the tensor with cartesian components [itex](\mathbf{a}\mathbf{b})_{ij} = a_ib_j[/itex]. This notation is introduced, for example, on pages 441ff of Boas (2nd edition). [itex]\nabla \mathbf{w}[/itex] is then the tensor with components [itex](\nabla \mathbf{w})_{ij} = \frac{\partial w_j}{\partial x_i} = \partial_i w_j[/itex].

This is about the point where vector notation should be abandoned in favour of suffices, as it becomes increasingly unclear which axes are involved in contractions.