Vector Divergence: Are the Expressions True?

[Arman777]

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 ?

 

[Abhishek11235]

Everything seems fine to me

 

[nrqed]

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 ?

I don't understand why you have a sum of derivatives of $J^m$. There is only one set of $J^m$, and each is a function of both position and time.

 

[Arman777]

Well yes that's the kind of the problem I am not sure how to express those things

 

[nrqed]

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 ?

Ok, the correct expressions are

EDIT: This is a better way to write it
$$\nabla' \cdot \vec{J} = \sum_{m=1}^3 \frac{\partial J^m(\vec{r},\vec{r'})}{\partial x'^m}$$
and
$$\nabla \cdot \vec{J} = \sum_{m=1}^3 \frac{\partial J^m(\vec{r},\vec{r}')}{\partial x^m} $$