Understanding the Laplacian of 1/r and the Dirac Delta Function Mathematical Foundations & SageMath Insights

Unveiling the Laplacian's Secrets: A Look at 1/r with SageMath

Engage & Explore!
Before we dive into the math, ask yourself:

• What does it mean when a function explodes to infinity at a point?

Leave your thoughts in the comments below. We'll revisit this after exploring the Dirac delta function!

Why Is the Laplacian of \( \frac{1}{r} \) Important?

The function \( \frac{1}{r} \) frequently appears in physics to describe potentials like gravity and electrostatics, which depend inversely on distance. Understanding its Laplacian reveals the nature of sources concentrated at singular points.

Mathematical Foundation: What Is the Laplacian and a Harmonic Function?

The Laplacian operator \( \Delta \) for a scalar function \( \phi(x,y,z) \) is defined as: \[ \Delta \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} \] A function is harmonic if its Laplacian is zero everywhere in its domain. Harmonic functions model steady-state or equilibrium situations in physics and engineering.

What Happens When You Take the Laplacian of \( \frac{1}{r} \)?

For \[ f(x,y,z) = \frac{1}{r}, \quad \text{where} \quad r = \sqrt{x^2 + y^2 + z^2} \] the Laplacian satisfies: \[ \Delta \left(\frac{1}{r} \right) = 0, \quad \text{for } r \neq 0 \] In plain terms: The function \( \frac{1}{r} \) is harmonic everywhere except at the origin.

What Happens at the Origin Where \( 𝑟=0 \) ?

The function \( \frac{1}{r} \) becomes infinite at the origin. This singularity means classical derivatives don’t apply there, so we use the framework of generalized functions (distributions).
A key result is: \[ \Delta \left(\frac{1}{r} \right) = -4\pi \delta(x) \] where \( \delta(x) \) is the three-dimensional Dirac delta function concentrated at the origin.
In simple words: The Laplacian of \( \frac{1}{r} \) acts like a point source at the origin, with strength \(−4\pi \).

How Does This Extend to 𝑛-Dimensions?

For \( n\geq3 \) , the fundamental solution behaves as: \[\Delta \left(\frac{1}{r^{n-2}} \right) = - (n-2) \Omega_n \delta(x) \] where \( \Omega_n \) is the surface area of the unit sphere in 𝑛-dimensions.

What Is Different in Two Dimensions?

In 2D, the fundamental solution changes to: \[ \ln \left( \frac{1}{r} \right)\] and its Laplacian is \[ \quad \Delta \ln \left( \frac{1}{r} \right) = -2\pi \delta(x) \] This reflects how flux spreads differently depending on dimension.

Verifying the Laplacian of \( \frac{1}{r} \) Using SageMath

SageMath can compute classical Laplacians away from singularities and verify numerical results but does not handle distributions symbolically.

      
# Load SageMath symbolic computation library
from sage.all import var, diff, sqrt

# Define symbolic variables for coordinates
x, y, z = var('x y z')

# Define the radial distance r from the origin
r_squared = x^2 + y^2 + z^2
r = sqrt(r_squared)

# Define the function f = 1/r
f = 1 / r

# Compute first derivatives of f with respect to x, y, z
df_dx = diff(f, x)  # ∂f/∂x
df_dy = diff(f, y)  # ∂f/∂y
df_dz = diff(f, z)  # ∂f/∂z

# Compute second derivatives of f
d2f_dx2 = diff(df_dx, x)  # ∂²f/∂x²
d2f_dy2 = diff(df_dy, y)  # ∂²f/∂y²
d2f_dz2 = diff(df_dz, z)  # ∂²f/∂z²

# Sum the second derivatives to get the Laplacian
laplacian_f = d2f_dx2 + d2f_dy2 + d2f_dz2

# Simplify the expression
laplacian_f_simplified = laplacian_f.simplify_full()

# Display the final result
print("Laplacian of 1/r:", laplacian_f_simplified)
	
    

💡 Try It Yourself! Want to explore the Laplacian in action?
Click below to run SageMath computations: You can copy and paste directly into here Run SageMath Code Here

The simplified Laplacian is zero for \( r \neq 0 \) , confirming \( \frac{1}{r} \) is harmonic away from the origin.

Visualizing the Singularity

Near the origin, \( \frac{1}{r} \) spikes sharply toward infinity, illustrating the singularity captured by the Dirac delta.

Conclusion: From Singularity to Insight

In this exploration, we've peeled back the layers of the Laplacian applied to \( \frac{1}{r} \) , a deceptively simple function with profound physical and mathematical significance. Here's what we uncovered:

• Harmonic Nature: \( \frac{1}{r} \) is harmonic everywhere except at the origin.
• The Power of Distributions: At the singularity, classical calculus breaks down — and distributions like the Dirac delta step in to make sense of it.
• Physical Insight: The delta function models a concentrated point source — a cornerstone concept in fields like electrostatics and gravitation.
• SageMath's Role: While not equipped for symbolic distribution theory, SageMath is a powerful tool for numerical validation and symbolic computation away from singularities.

This journey highlights the beauty of blending mathematical theory, computational tools, and physical intuition. Whether you're a student, researcher, or curious thinker, mastering concepts like these opens new doors in mathematical physics and applied analysis.