- #1

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- Homework Statement
- Derive the Identitiy using Einstein Summation Notation

- Relevant Equations
- $$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2}) = ?$$

I have an identity

$$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2})$$

which should give us

$$3(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m}$$

But I have to derive it using the Einstein summation notation.

How can I approach this problem to simplify things ?

Should I do something like ##\vec{k}=\vec{m} \times \hat{r}## ? and then

$$\vec{\nabla} \times (\frac{\vec{k}}{r^2}) = \frac{r^2 \nabla \times \vec{k} - \nabla(r^2) \times \vec{k}}{ r^4} $$ ? But it seems like things getting more complicated this way.

$$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2})$$

which should give us

$$3(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m}$$

But I have to derive it using the Einstein summation notation.

How can I approach this problem to simplify things ?

Should I do something like ##\vec{k}=\vec{m} \times \hat{r}## ? and then

$$\vec{\nabla} \times (\frac{\vec{k}}{r^2}) = \frac{r^2 \nabla \times \vec{k} - \nabla(r^2) \times \vec{k}}{ r^4} $$ ? But it seems like things getting more complicated this way.