# Dirac delta function definition

• latentcorpse
In summary: Thus, the two expressions are equivalent.In summary, the Dirac delta function can be used to express a 3D integral in terms of a 2D integral, which simplifies the calculation. This can be seen in the application of the delta function in the calculation of \mathbf{\nabla} \wedge \mathbf{B}(\mathbf{r}). The two forms of the delta function, \delta(\mathbf{r}-\mathbf{r'}) and \delta(\mathbf{r'}-\mathbf{r}), are equivalent in this case because the delta function is symmetric about the spike. This allows for the simplification of the integral and leads to the desired result of \mu
latentcorpse
By definition of the Dirac delta function, we have:

$\int f(x) \delta(x-a) dx=f(a)$

This is fair enough. But in ym notes there is a step that goes like the following:

$\mathbf{\nabla} \wedge \mathbf{B}(\mathbf{r})=-\frac{\mu_0}{4 \pi} \int_V dV' \nabla^2(\frac{1}{|\mathbf{r}-\mathbf{r'}|}) \mathbf{J}(\mathbf{r'}) = \mu_0 \mathbf{J}(\mathbf{r})$

where we have used that $\nabla^2(\frac{1}{|\mathbf{r}-\mathbf{r'}|}) =-4 \pi \delta(\mathbf{r}-\mathbf{r'})$

clearly the minus signs and the $4 \pi$'s cancel so it's now just

$\mathbf{\nabla} \wedge \mathbf{B}(\mathbf{r})=\mu_0 \int_V dV' \mathbf{J}(\mathbf{r'}) \delta(\mathbf{r}-\mathbf{r'})$

i don't see how that goes from there to give $\mu_0 \mathbf{J}(\mathbf{r})$ as we are integrating with respect to $V'$ are we not? and so i would assume that the delta function would need to be of the form $\delta(\mathbf{r'}-\mathbf{r})$ in order to give the desired answer.

the onnly explanation i can come up with is that $\delta(\mathbf{r'}-\mathbf{r})=\delta(\mathbf{r}-\mathbf{r'})$ since the delta function is symmetric about the spike. however the spike would be at different positions in these two cases. I'm kind of lost-any advice?

latentcorpse said:
the onnly explanation i can come up with is that $\delta(\mathbf{r'}-\mathbf{r})=\delta(\mathbf{r}-\mathbf{r'})$ since the delta function is symmetric about the spike.

Yes, that's true.

however the spike would be at different positions in these two cases. I'm kind of lost-any advice?

No, the delta function has a spike whenever it's argument is zero. In both cases this occurs at $r=r'$.

## What is the Dirac delta function?

The Dirac delta function, commonly denoted as δ(x), is a mathematical function that is used to represent a point mass or impulse at a specific location. It has a value of 0 everywhere except at the point of interest, where it has an infinite value. It is often used in physics and engineering to model idealized point sources or events.

## How is the Dirac delta function defined mathematically?

The Dirac delta function is defined as follows: δ(x) = 0 for all x ≠ 0, and δ(0) = ∞. It also has the property that ∫δ(x)dx = 1, meaning that the area under the curve of the function is equal to 1. It is a distribution, rather than a traditional function, as it does not have a well-defined value at every point.

## What is the physical interpretation of the Dirac delta function?

The Dirac delta function can be thought of as a spike or impulse at a specific point in space or time. It is often used to represent point sources, such as a point charge in electromagnetism or a point mass in mechanics. It can also be used to model sudden or instantaneous events, such as an explosion or a collision.

## How is the Dirac delta function used in mathematical calculations?

The Dirac delta function is used in a variety of mathematical calculations, particularly in integration and differential equations. It can be used to simplify mathematical expressions and make calculations more efficient. It is also commonly used in Fourier analysis and signal processing.

## What are some real-world applications of the Dirac delta function?

The Dirac delta function has many applications in physics, engineering, and other fields. It is used in quantum mechanics to represent the position and momentum of particles, in signal processing to model idealized signals, and in probability theory to represent impulses in stochastic processes. It is also used in image processing, fluid mechanics, and many other areas of science and engineering.

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