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

[Brix12]

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!

 

[PeroK]

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.

 

[robphy]

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

 

[pasmith]

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

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.