Electrodynamics: Derivatives involving Retarded-Time

[WWCY]

Hi all,

I have ran into some mathematical confusion when studying the aforementioned topic. The expression for retarded time is given as
$$t_R = t - R/c$$
$R = | \vec{r} - \vec{r'} |$, where $\vec{r}$ represents the point of evaluation and $\vec{r'}$ represents the source position. I rewrite the above equation into
$$x_i = x'_i + \text{terms independent of x' and x}$$
where $x_i$ is a cartesian coordinate. If I had the following derivative $\partial _{x_i}$, could I then say this?
$$\partial _{x_i} = \frac{ \partial }{ \partial x'_i} \frac{ \partial x'_i}{ \partial x_i} = \partial _{x'_i}$$

If so, there is the following identity
$$\nabla(1/R) = - \nabla ' (1/R)$$
where the prime means that the derivative is with respect to source coordinates $x' _i$

Suppose I start from the RHS:
$$\nabla (1/R) = \sum_i \partial_ {x_i} (\frac{1}{R}) \hat{x_i} = \sum_i \partial_ {x' _i} (\frac{1}{R}) \hat{x_i} = \nabla' (1/R)$$

which is clearly wrong. What have I done wrongly?

Thanks in advance!

 

[Delta2]

Your mistake is that you consider the dependency equation $x_i=x'_i+C$ between the field coordinates and the source coordinates but there can be no such dependency, we treat all field variables $x_i$ and all source variable $x'_i$ as independent from each other when proving the identity $\nabla (1/R)=-\nabla'(1/R)$.