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- Homework Statement:
- Show the identity ##\vec{\nabla}(\vec{r} \cdot \vec{u}) = \vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{L} \times \vec{u})##, ##\vec{L} = \frac{\vec{r}}{i} \times \vec{\nabla}##

- Relevant Equations:
- ##\vec{\nabla}(\vec{r} \cdot \vec{u}) = \vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{L} \times \vec{u})##, ##\vec{L} = \frac{\vec{r}}{i} \times \vec{\nabla}##

First of all, sorry for the title I don't know the name of this formula and that's part of the problem, I can't find anything on google.

I have to show the identity above. Here's what I did. I don't know if this is correct so far.

##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{L} \times \vec{u})##

= ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\frac{\vec{r}}{i} \times \vec{\nabla} \times \vec{u})##

= ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{\nabla}(\frac{\vec{r}}{i} \cdot \vec{u}) - \vec{u}(\frac{\vec{r}}{i} \cdot \vec{\nabla}))##

= ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + \vec{\nabla}(\vec{r} \cdot \vec{u}) - \vec{u}(\vec{r} \cdot \vec{\nabla})##

Is there a specific name for this identity?

I have to show the identity above. Here's what I did. I don't know if this is correct so far.

##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{L} \times \vec{u})##

= ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\frac{\vec{r}}{i} \times \vec{\nabla} \times \vec{u})##

= ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{\nabla}(\frac{\vec{r}}{i} \cdot \vec{u}) - \vec{u}(\frac{\vec{r}}{i} \cdot \vec{\nabla}))##

= ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + \vec{\nabla}(\vec{r} \cdot \vec{u}) - \vec{u}(\vec{r} \cdot \vec{\nabla})##

Is there a specific name for this identity?

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