1 スカラーポテンシャルの座標微分

$$\deL{x}\phi\xt = \frac{1}{4\pi\vepsilon_0} \Int dt' \, \deL{x}\le... ... dt' \, \delta\left(h(\vx,t,t')\right)\deL{x} \left[\frac{q}{R(\vx,t')}\right]$$

であるが、それぞれの項にについて計算を次のように行う。

$$\hbox{（第弐項）} =\frac{1}{4\pi \vepsilon_0} \Int dt' \, \delta\left(h(\vx,t,t')\r... ...ilon_0} \Int dt' \, \delta\left(h(\vx,t,t')\right) \frac{q(\vn)_x}{R^2(\vx,t')}$$

$$=-\frac{q}{4\pi \vepsilon_0}\frac{1}{\dfrac{d}{dt'} \left( t' -t ... ...t')\cdot \bm{\beta}(t')} \frac{(\vn)_x}{R^2(\vx,t')}\Bigg\vert _{h(\vx,t,t')=0}$$

$$=-\frac{q}{4\pi \vepsilon_0}\frac{1}{\kappa(t')} \frac{(\vn)_x}{R^2(\vx,t')}\Bigg\vert _{h(\vx,t,t')=0}$$

$$\hbox{（第壱項）} =\frac{1}{4\pi\vepsilon_0} \Int dt' \, \deL{x}\left[ \delta\left(... ... +\frac{\left\vert\vx-\vx'(t')\right\vert}{c} \right)\right]\frac{1}{R(\vx,t')}$$

$$= \frac{q}{4\pi\vepsilon_0} \frac{1}{c} \Int dy \, \di{t'}{y} \fr... ... \deL{y} \delta(t-y),\qquad y= t' + \frac{\left\vert\vx-\vx'(t')\right\vert}{c}$$

$$=-\frac{q}{4\pi\vepsilon_0} \frac{1}{c} \Int dy \, \deL{y}\left( ... ...} \frac{(\vn)_x}{R(\vx,t')} \right)\delta(t-y) \qquad \because\,\hbox{部分積分}$$

$$=-\frac{q}{4\pi\vepsilon_0} \frac{1}{c} \deL{t} \left[\di{t'}{t} ... ...'}{t}\deL{t'} \left[\di{t'}{t} \frac{(\vn)_x}{R(\vx,t')}\right]_{h(\vx,t,t')=0}$$

$$= -\frac{q}{4\pi\vepsilon_0} \frac{1}{c} \frac{1}{\kappa(t')}\deL... ...]_{h(\vx,t,t')=0} \qquad \because\, \di{t'}{t}=\del{t'}{t}=\frac{1}{\kappa(t')}$$

よって、

$$\deL{x} \phi\xt =-\frac{q}{4\pi\vepsilon_0} \frac{1}{c} \frac{1}{... ...n_0}\frac{1}{\kappa(t')} \frac{(\vn)_x}{R^2(\vx,t')}\Bigg\vert _{h(\vx,t,t')=0}$$ (33)

となる。