In the book Advanced Classical Electromagnetism by Robert Wald, it was shown if $\psi$ satisfy the homogeneous wave equation $$\square\psi=0$$ then we have $$\psi\left(x^{\mu}\right)=-\frac{1}{c} \int_{x^{\prime 0}=0}\left[\psi\left(x^{\prime \mu}\right) \partial_{0}^{\prime} G_{\mathrm{ret}}\left(x^{\mu}, x^{\prime \mu}\right)-G_{\mathrm{ret}}\left(x^{\mu}, x^{\prime \mu}\right) \partial_{0}^{\prime} \psi\left(x^{\prime \mu}\right)\right] d^{3} x^{\prime}\tag{5.133}$$ where the retarded Green function is defined as $$G_{\text {ret }}\left(t, x ; t^{\prime}, x^{\prime}\right)= \begin{cases}0, & \text { for } t<t^{\prime} \\ \frac{\delta\left(t-t^{\prime}-\left|x-x^{\prime}\right| / c\right)}{4 \pi\left|x-x^{\prime}\right|}, & \text { for } t>t^{\prime}\end{cases}\tag{5.50}$$ but in the next line, he suddenly has $$\psi(t, \boldsymbol{x})=\frac{1}{4 \pi} \int_{S^{\prime}}\left[\frac{1}{r^{\prime 2}} \psi\left(\theta^{\prime}, \varphi^{\prime}\right)+\frac{1}{r^{\prime}}\left(\frac{1}{c} \frac{\partial \psi}{\partial t}\left(\theta^{\prime}, \varphi^{\prime}\right)+\hat{\boldsymbol{r}}^{\prime} \cdot \nabla \psi\left(\theta^{\prime}, \varphi^{\prime}\right)\right)\right] r^{\prime 2} \sin \theta^{\prime} d \theta^{\prime} d \varphi^{\prime}\tag{5.134}$$ for $t>0$. I find this step particularly hard to follow, here are some of my questions:
- Can we assume that the usual rule defining derivative of delta function (distribution) continue to hold in this case, such that $\delta' f=-\delta f'$?
- If so, where does the first and third term of the integrand of (5.134) come from? Since the derivative is taken with respect to the zeroth (i.e. time) component in (5.133), I feel there shouldn't be a gradient term in (5.134). Also, I wonder why the first term scale as $1/r'^2$...