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- Homework Statement
- ##\nabla \cdot \vec{J}## and ##\nabla' \cdot \vec{J}$## ?
- Relevant Equations
- 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
$$\nabla = \frac{\partial}{\partial x}\vec{i} +\frac{\partial}{\partial y}\vec{j} +\frac{\partial}{\partial z}\vec{k}$$
Do I have to write something like,
$$\nabla' \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x'^m} + \frac{\partial J^m(t_r)}{\partial x'^m}$$
$$\nabla \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x^m} + \frac{\partial J^m(t_r)}{\partial x^m} = \frac{\partial J^m(t_r)}{\partial x^m}$$
Are these expressions true ?
$$\nabla' \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x'^m} + \frac{\partial J^m(t_r)}{\partial x'^m}$$
$$\nabla \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x^m} + \frac{\partial J^m(t_r)}{\partial x^m} = \frac{\partial J^m(t_r)}{\partial x^m}$$
Are these expressions true ?