The Origin of Singularities: Unpacking the Laplacian of 1/r & the Birth of the Delta Function!

The Origin Story of a Singularity: Unpacking the Laplacian of 1/r (and the Birth of the Delta Function!)

Ever wondered what happens when smooth math slams into an infinite wall? What if a function tries to break free from its well-behaved nature... and creates a mathematical black hole at its core?
In our last journey, we brushed against the edges of functions that "break," hinting at something deeper: singular functionals. Today, we dive headfirst into the most iconic of them all — the mysterious behavior of \( 1/r\) under the Laplacian.
Get ready for a revelation: how this innocent-looking function hides the secret origin of the Dirac delta function, a tool so powerful it reshaped fields from quantum mechanics to engineering.

The Curious Case of 1/r: A Harmonic Hero with a Hidden Flaw

Recall the 3D radial distance:\[ \sqrt{(x_1^2+ x_2^2 + x_3^2} \]The function 1/r is everywhere in physics:

• It's the gravitational potential around a star.
• It's the electric potential of a point charge.
• It's a fundamental solution of Laplace’s equation.

It’s harmonic (meaning \(\Delta \left(\frac{1}{r} \right) = 0\) everywhere except at the origin.
But what happens at the origin?
That’s where things get spicy. At \( \quad r = 0, \quad \frac{1}{r} \) explodes into a singularity. It’s like the function folds space into an infinite spike. How do we make sense of this in mathematics?

The Detective Work: Laplacian as a Generalized Function

To investigate, we enter the world of generalized functions (distributions), where we don’t evaluate functions directly — we test them using smooth “probe” functions \(\) The Laplacian of \(\frac{1}{r} \) becomes:\[(\Delta (1/r), \varphi) = \left( \frac{1}{r}, \Delta \varphi \right) = \iiint_{\mathbb{R}^3} \frac{1}{r} \Delta \varphi \, dV \]But wait — we can’t just integrate over all of space. That singularity at the origin is a no-go zone. So we cut out a tiny ball of radius 𝜖, do the integration outside it, and then shrink ϵ→0.

Green's Theorem: The Mathematical Microscope

Using Green’s identity on a ball with a spherical hole at the center: \[ \iiint_{r \geq \epsilon} \frac{1}{r} \Delta \varphi \, dV = \iiint_{r \geq \epsilon} \varphi \Delta \left(\frac{1}{r} \right) \, dV - \iint_{S_{\epsilon}} \left( \varphi \frac{\partial}{\partial n} \left(\frac{1}{r} \right) - \frac{1}{r} \frac{\partial \varphi}{\partial n} \right) \, dS\] Breakdown:

• Volume Integral: Vanishes because \(\Delta \left( \frac{1}{r} \right) = 0 \) for \( r \neq 0.\)
• First Surface Term: \[\frac{\partial}{\partial n} \left( \frac{1}{r} \right) = -\frac{1}{r^2} \Rightarrow \iint_{S_{\epsilon}} \varphi \cdot \frac{1}{\epsilon^2} \, dS \to 4\pi \varphi(0)\]
• Second Surface Term: Goes to 0 as ϵ→0
  Result: \[ (\Delta (1/r), \varphi) = 4\pi \varphi(0) = 4\pi (\delta(x), \varphi) \]

The Grand Reveal

Thus, \[ \Delta (1/r) = 4\pi \varphi(x) \] This means all the "action" is concentrated at the origin. The Laplacian detects the singularity and encodes it as a delta function.

SageMath Spotlight: Visualizing the Delta’s Ghost

You can’t graph an infinite spike — but you can approximate it! Try this interactive visualization with SageMath:

      
# Define symbolic variables
x = var('x')
epsilon = var('epsilon')

def approx_delta(x, epsilon):
    return (1/(epsilon * sqrt(pi))) * exp(-x^2 / epsilon^2)

# Plot for different epsilons
p1 = plot(approx_delta(x, 0.5), (x, -2, 2), color='blue', legend_label='ε = 0.5')
p2 = plot(approx_delta(x, 0.1), (x, -2, 2), color='red', legend_label='ε = 0.1')
p3 = plot(approx_delta(x, 0.05), (x, -2, 2), color='green', legend_label='ε = 0.05')

show(p1 + p2 + p3, title="Approximating the Dirac Delta Function")

# Integral check
test_epsilon = 0.01
integral_val = integrate(approx_delta(x, test_epsilon), x, -infinity, infinity)
print(f"Integral ≈ {integral_val.n()}")
	
    

💡 Try It Yourself! Now You can copy and paste directly into here Run SageMath Code Here

Challenge: Set \(epsilon = 0.001\) and rerun it. What happens to the peak? Does the area stay close to 1?

Why It Matters: Singularities in the Wild

Singularities aren't just math tricks — they model real, jaw-dropping physics:

• Electrostatics: A point charge’s potential is 1/𝑟 , and the charge distribution is 𝛿(𝑥).
• Fluid Dynamics: Shockwaves create discontinuities in pressure and flow — modeled as singularities.
• Material Science: Cracks concentrate stress into tiny zones — mathematically, a singularity.
• Black Holes: General relativity predicts infinite curvature — singularities of spacetime itself!
• Quantum Mechanics: Wavefunction collapse = instant transition — modeled using delta-like behavior.
• Phase Transitions: Supercooling or nucleation involves sudden changes — modeled with singular terms.

The Dirac delta is the mathematical embodiment of everything happening at a single point — whether it's a charge, a crack, a collapse, or a cosmic mystery.

Beyond 3D: A Unified Theory of Point Sources

This isn’t just about 3D. In \( n \geq 3 \)dimensions:\[ \quad \Delta \left( \frac{1}{r^{n-2}} \right) = - (n-2) \Omega_n \delta(x)\] For n = 3, we recover: \[ \quad \Delta \left( \frac{1}{r} \right) = -4\pi \delta(x) \] In 2D, we switch to: \[ \quad \Delta \left( \ln \left(\frac{1}{r} \right) \right) = -2\pi \delta(x) \]

These are the fundamental solutions — the “fingerprints” of localized sources in space.

Where We’re Headed: Differentiation Without Limits

Coming soon: How to treat singular objects like delta functions and even 1/𝑟 as if they were smooth, by upgrading our idea of differentiation itself. This will open doors to:

• Solving PDEs with singular sources
• Creating Green's functions
• Making physics and engineering models more precise

Your Turn: Think Like a Physicist

Here’s your challenge:

If \( \Delta( 1/r)=4\pi\delta(x) \) and 1/r is the potential of a point charge, how does this deepen your understanding of the delta function as a source? Can you feel why the Laplacian acts like a “charge detector”? What happens when there are multiple sources — or a continuous distribution?

Share your thoughts, insights, or even code experiments in the comments. Let’s explore the singular frontier together.