Vector Fields

In single-variable calculus, we map a number to a number ($x \to y$). In Vector Calculus, we map a position to a vector ($\mathbf{r} \to \mathbf{V}$). This represents physical phenomena like wind speed at every point in a city, or the force of gravity at every point in the solar system.

The Gradient: The Direction of Maximum Ascent

Given a scalar field $f(x, y)$ (like the altitude of a mountain), the Gradient $\nabla f$ is a vector field that points in the direction of the steepest uphill slope.

$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$

The magnitude of the gradient tells you how steep the slope is.

Divergence: Sources and Sinks

The Divergence ($\nabla \cdot \mathbf{V}$) of a vector field measures how much the “fluid” is expanding or compressing at a point.

• Positive Divergence: The point is a Source (fluid is created/flowing out).
• Negative Divergence: The point is a Sink (fluid is destroyed/flowing in).

Curl: The Rotation of the Field

The Curl ($\nabla \times \mathbf{V}$) measures the tendency of the field to rotate around a point. If you placed a tiny paddlewheel in the field, the curl tells you how fast and in what direction it would spin.

import numpy as np
import matplotlib.pyplot as plt

# Define a grid of points
x, y = np.meshgrid(np.linspace(-2, 2, 10), np.linspace(-2, 2, 10))

# Example Vector Field: Rotational field (V = [-y, x])
u = -y
v = x

print("Visualizing a rotational field (Curl > 0):")
# In a real environment, we'd use plt.quiver(x, y, u, v)
# For now, let's look at the vectors at a few points
for i in [0, 5, 9]:
  for j in [0, 5, 9]:
      print(f"Point ({x[i,j]:.1f}, {y[i,j]:.1f}) -> Vector ({u[i,j]:.1f}, {v[i,j]:.1f})")

Conservative Fields

A vector field is Conservative if it is the gradient of some scalar function ($V = \nabla f$). In physics, this means the work done moving between two points is independent of the path taken. Gravity and electric fields are conservative; friction is not.

Exercises