I am doing EMT and I am trying to calculate the divergence of this current density given as,

$$\vec{J}(\vec{r}', t_r) = \vec{J}(\vec{r}', t - \frac{|\vec{r}-\vec{r}'|}{c})$$

for $\vec{r} = (x,y,z)$ and $\vec{r'} = (x',y',z')$ Now We have two divergence operator,

$$\nabla' = \frac{\partial}{\partial x'}\vec{i} +\frac{\partial}{\partial y'}\vec{j} +\frac{\partial}{\partial z'}\vec{k}$$ and

$$\nabla = \frac{\partial}{\partial x}\vec{i} +\frac{\partial}{\partial y}\vec{j} +\frac{\partial}{\partial z}\vec{k}$$

$\nabla \cdot \vec{J}$ and $\nabla' \cdot \vec{J}$ ?

How can I calculate these expressions ?

I guess

$$\nabla \cdot \vec{J} = \frac{\partial J^m}{\partial x^m} = \frac{\partial J^m}{\partial t_r}\frac{\partial t_r}{\partial x^m}$$

But I dont know how to express $\nabla' \cdot \vec{J}$


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