Calculus 11: Divergence, Curl, and Laplacian — Diffusion, Heat, and Curvature

“The Laplacian captures the ‘flow of flow’ — the ultimate second-derivative. In physics: heat. In ML: smoothness, diffusion, and curvature.”

1 · Key Operators: Divergence, Curl, Laplacian

For a 2D vector field $\mathbf{F}(x, y) = (F_x, F_y)$:

• Divergence:

  $$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}$$

  (How much “outflow” from a point.)

• Curl:

  $$\nabla \times \mathbf{F} = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}$$

  (Local “rotation” of the field.)

• Laplacian: For scalar field $f(x, y)$:

  $$\Delta f = \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$$

  (Sum of 2nd derivatives; a “curvature” operator.)

2 · Why ML Engineers Care

3 · Demo ① — The Heat Equation on a Grid

The heat equation in 2D:

where $D$ is the diffusion constant, and $\Delta u$ is the Laplacian.

Discretized: We can approximate the Laplacian with a convolution kernel:

Python Demo: Simulate Heat Diffusion

# calc-11-div-curl-laplace/heat_equation_demo.py
import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import convolve

N = 64
D = 0.15   # diffusion constant
steps = 80

# Laplacian kernel for 2D grid
kernel = np.array([[0, 1, 0],
                   [1, -4, 1],
                   [0, 1, 0]])

# Initial "hot spot" in center
u = np.zeros((N, N))
u[N//2, N//2] = 10.0

# For plotting
fig, axs = plt.subplots(2, 4, figsize=(12, 6), sharex=True, sharey=True)
plot_steps = np.linspace(0, steps-1, 8, dtype=int)

for s in range(steps):
    lap = convolve(u, kernel, mode="constant")
    u += D * lap
    if s in plot_steps:
        ax = axs.flat[list(plot_steps).index(s)]
        im = ax.imshow(u, vmin=0, vmax=10, cmap="hot")
        ax.set_title(f"step {s}")

plt.colorbar(im, ax=axs.ravel().tolist(), shrink=0.85)
plt.suptitle("2D Heat Equation Evolution (Diffusion of Hot Spot)")
plt.tight_layout(rect=[0,0,1,0.96])
plt.show()

4 · Connection to ML

• Diffusion models: Recent generative models (DDPMs, etc.) literally simulate heat/diffusion in data space.
• Graph Laplacian: On graphs, the Laplacian encodes “smoothness” and is used for clustering and semi-supervised learning.
• Regularization: Penalizing $\Delta f$ encourages smooth functions.

5 · Exercises

1. Divergence and Curl on a Grid: For $\mathbf{F}(x, y) = [y, -x]$, compute divergence and curl at each grid point in $[-2,2]^2$, and plot them as images.
2. Laplacian of a Gaussian: Compute and plot $\Delta f$ for $f(x, y) = \exp(-x^2 - y^2)$. Where is it most negative?
3. Multiple Hot Spots: Initialize $u$ with three “hot spots.” Simulate the heat equation and show how the blobs merge.
4. Graph Laplacian: Construct a 6-node ring graph Laplacian. Simulate diffusion of an initial delta on one node for 10 steps and plot the value on each node at each step.

Put solutions in calc-11-div-curl-laplace/ and tag v0.1.

Next: Calculus 12 — ODEs, Gradient Flows, and Neural Dynamics.