Notation regarding deltafunction derivatives

[JustinLevy]

Consider the current of some point particles
$$\vec{j}(\vec{r},t) = \sum_\alpha q_\alpha \dot{\vec{r}}_\alpha(t) \delta^3(\vec{r}_\alpha(t) - \vec{r})$$

If I wanted to take the time derivative of this, what would be the best way to write it notationally?


I assume this is clear enough:
$$\dot{\vec{j}}(\vec{r},t) = \sum_\alpha [ q_\alpha \ddot{\vec{r}}_\alpha(t) \delta^3(\vec{r}_\alpha(t) - \vec{r}) + q_\alpha \dot{\vec{r}}_\alpha(t) \frac{d}{dt}\delta^3(\vec{r}_\alpha(t) - \vec{r}) ]$$

It is my understanding that the derivative of a delta "function" can be written:
$$\frac{d}{dt} \delta(t) = -\frac{1}{t}\delta(t)$$
which I assume means I can write (where $a$ is some constant)
$$\frac{d}{dt} \delta(t-a) = -\frac{1}{t-a}\delta(t-a)$$
correct?

So I could manipulate that last term like this?
$$\frac{d}{dt}\delta^3(\vec{s}(t) - \vec{r}) = \frac{d}{dt}\delta(s_x(t) - x)\delta(s_y(t) - y)\delta(s_z(t) - z) = \frac{d}{dt} \sum_{i} \frac{1}{|\dot{s}_x(t_i)|}\frac{1}{|\dot{s}_y(t_i)|}\frac{1}{|\dot{s}_z(t_i)|} \delta(s_x(t_i) - x)\delta(s_y(t_i) - y)\delta(s_z(t_i) - z)$$
where $$t_i$$ is a time satisfying [tex]\vec{s}(t_i) = \vec{r}.[/itex]

Then I have:
$$\frac{d}{dt}\delta^3(\vec{s}(t) - \vec{r}) = \sum_{i} \left(\frac{1}{s_x(t_i) - x} + \frac{1}{s_y(t_i) - y} + \frac{1}{s_z(t_i) - z}\right)\frac{1}{|\dot{s}_x(t_i)|}\frac{1}{|\dot{s}_y(t_i)|}\frac{1}{|\dot{s}_z(t_i)|} \delta(s_x(t_i) - x)\delta(s_y(t_i) - y)\delta(s_z(t_i) - z)$$


First of all, is that correct?
Second of all, is there a better notational way to write that?

 

[jvicens]

Yes, your manipulation of the delta function is correct. As for a better notational way to write it, you could use the chain rule for derivatives and write it as:

$$\frac{d}{dt}\delta^3(\vec{s}(t) - \vec{r}) = \sum_i \frac{1}{|\nabla \vec{s}(t_i)|} \frac{\partial \vec{s}(t_i)}{\partial t} \cdot \nabla \delta^3(\vec{s}(t_i) - \vec{r})$$

where $\nabla$ is the gradient operator. This notation may be more compact and easier to understand for someone familiar with vector calculus. However, if your audience is not familiar with vector calculus, it may be better to stick with your original notation.