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

  • #1
Brix12
1
0
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!
 
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  • #2
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.
 
  • #3
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
 
  • #4
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.
 

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