# Notation regarding deltafunction derivatives

• JustinLevy
In summary, there are different ways to notate the derivative of a delta function, including using the chain rule and the gradient operator, but it is important to consider your audience when choosing the most appropriate notation.
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 $$\vec{s}(t_i) = \vec{r}.[/itex] Then I have: [tex]\frac{d}{dt}\delta^3(\vec{s}(t) - \vec{r}) = \sum_{i} \left(\frac{1}{s_x(t_i) - x} + \frac{1}{s_y(t_i) - y} + \frac{1}{s_z(t_i) - z}\right)\frac{1}{|\dot{s}_x(t_i)|}\frac{1}{|\dot{s}_y(t_i)|}\frac{1}{|\dot{s}_z(t_i)|} \delta(s_x(t_i) - x)\delta(s_y(t_i) - y)\delta(s_z(t_i) - z)$$

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

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.

## 1. What is the notation used for delta function derivatives?

The notation typically used for delta function derivatives is $\delta'$ or $\delta^{(1)}$.

## 2. How do you represent higher order derivatives of the delta function?

Higher order derivatives of the delta function can be represented using the notation $\delta^{(n)}$, where n is the order of the derivative.

## 3. What is the purpose of using delta function derivatives?

Delta function derivatives are commonly used in mathematical and physical equations to represent point sources or impulses.

## 4. Are there any properties of delta function derivatives?

Yes, there are several properties of delta function derivatives including the derivative of a delta function is equal to a scaled delta function, and the integral of a delta function derivative is equal to zero.

## 5. Can delta function derivatives be used in multivariate functions?

Yes, delta function derivatives can be used in multivariate functions by taking the derivative with respect to each variable separately.

Replies
5
Views
1K
Replies
7
Views
1K
Replies
0
Views
713
Replies
1
Views
1K
Replies
3
Views
521
Replies
3
Views
974
Replies
1
Views
2K
Replies
6
Views
3K
Replies
0
Views
806
Replies
2
Views
936