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Physics
Classical Physics
Electromagnetism
Electrodynamics: Derivatives involving Retarded-Time
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[QUOTE="WWCY, post: 6130599, member: 608494"] 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! [/QUOTE]
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Forums
Physics
Classical Physics
Electromagnetism
Electrodynamics: Derivatives involving Retarded-Time
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