5: Dirac delta function

Section 1: Vector Calculus - 5: Dirac Delta Function

The Dirac delta function, denoted $\delta(\mathbf{r} - \mathbf{r}_0)$ or $\delta(x - x_0)$ in 1D, is a crucial generalized function (or distribution) in physics and vector calculus. It is rigorously defined by its integral properties rather than pointwise values. Its primary purpose is to model infinitely concentrated sources, such as point charges or masses.

Core Definition & Sifting Property:
In one dimension, $\delta(x - x_0)$ satisfies:

1. $\delta(x - x_0) = 0$ for all $x \neq x_0$
2. $\int \delta(x - x_0) dx = 1$ over any interval containing $x_0$.
   Its most vital feature is the sifting property:
   $\int f(x) \delta(x - x_0) dx = f(x_0)$
   This "picks out" the value of a test function $f(x)$ at the point $x_0$.

Three-Dimensional Delta Function:
For 3D space (relevant to electromagnetism), the delta function is defined as:
$\delta^3(\mathbf{r} - \mathbf{r}_0) = \delta(x - x_0) \delta(y - y_0) \delta(z - z_0)$
in Cartesian coordinates. It satisfies:
$\int f(\mathbf{r}) \delta^3(\mathbf{r} - \mathbf{r}_0) dV = f(\mathbf{r}_0)$
where $dV$ is the volume element. The integral is 1 if the volume contains $\mathbf{r}_0$, and 0 otherwise.

Key Properties:

• Scaling: $\delta(kx) = \frac{1}{|k|} \delta(x)$
• Coordinate Transformation: Under a change of variables, $\delta^3(\mathbf{r} - \mathbf{r}_0)$ transforms with a factor of $1/|J|$, where $J$ is the Jacobian determinant of the transformation, ensuring the sifting property holds.

Connection to Vector Calculus & Divergence Theorem:
The delta function appears fundamentally when taking divergences of fields with singularities. A critical result, derived using the divergence theorem, is:
$\nabla \cdot \left( \frac{\hat{\mathbf{r}}}{r^2} \right) = 4\pi \delta^3(\mathbf{r})$
where $\hat{\mathbf{r}}$ is the unit radial vector and $r = |\mathbf{r}|$. This shows the divergence vanishes everywhere except at the origin, where it represents a point source. Similarly, the Laplacian of $1/r$ is:
$\nabla^2 \left( \frac{1}{r} \right) = -4\pi \delta^3(\mathbf{r})$
These expressions are essential for solving Poisson's equation ($\nabla^2\phi = -\rho/\epsilon_0$) for point charges, where $\rho(\mathbf{r}) = q \delta^3(\mathbf{r} - \mathbf{r}_0)$.