Gradient of a time-dependent function

[Sturk200]

Hi!

I am struggling with what I think is probably a fairly simple step in Landau & Lifshitz derivation of the fields from the Lienard-Wiechert potential. We have the potential in terms of a primed set of coordinates but the fields are defined in terms of derivatives with respect to unprimed coordinates; the two are related by

$$t'+{R(t')\over{c}} = t$$

where R is the distance from the point charge to the field point at time t' and time t is the moment of observation.

The step that I am having trouble with is in finding an expression for the gradient of t'. Landau has:

$$\nabla t' = -\frac{1}{c}\nabla R(t') = -\frac{1}{c} \bigg(\frac{\partial R}{\partial t'} \nabla t' + \frac{\textbf{R}}{R}\bigg)$$

The first equality obviously follows from the equation above. The second equality is where I am stumped. I would think that it should be simply

$$\nabla R(t') = \frac{\partial R}{\partial t'} \nabla t'$$.

Does anyone know where that extra unit vector term comes from?

Thanks!

 

[RUber]

It looks to me like that came from the unprimed t in your original equation.
$t' = -\frac{R(t')}{c} +t$
so
$\nabla t' = -\nabla \frac{R(t')}{c} +\nabla t$
Is it reasonable to write $\nabla t = \frac{-c}{R}\mathbf{R}$?

 

[Sturk200]

It looks to me like that came from the unprimed t in your original equation.
$t' = -\frac{R(t')}{c} +t$
so
$\nabla t' = -\nabla \frac{R(t')}{c} +\nabla t$
Is it reasonable to write $\nabla t = \frac{-c}{R}\mathbf{R}$?

I don't think that makes sense, since it is not consistent with the first equality:

$$\nabla t' = -\frac{1}{c}\nabla R(t')$$.