Introduction to Vector Fields¶

Part 0 of the GRL Field Series — Gentle Introduction

Welcome! This notebook provides a gentle introduction to vector fields with real-world examples and intuition-building.

Learning Objectives¶

By the end of this notebook, you'll understand:

What is a vector field? (arrows at each point in space)
Gradient fields — following "uphill" directions

Prerequisites: Basic calculus (derivatives), linear algebra (vectors)

Time: ~10-15 minutes

Why This Matters for GRL¶

Vector fields are the stepping stone to functional fields. Once you understand how arrows at each point create a field, you'll be ready to understand how functions at each point create GRL's reinforcement field!

Progression:

Introduction (this notebook) ← You are here
    ↓
Classical Vector Fields (Notebook 1)
    ↓
Functional Fields (Notebook 2)
    ↓
GRL Reinforcement Fields (Notebook 3)

Next: For a complete treatment including rotational fields, superposition, and trajectories, see 01_classical_vector_fields.ipynb.

In [ ]:

Copied!





# Setup: Import libraries
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
import seaborn as sns

# Set visualization style
sns.set_theme(style='whitegrid', context='notebook')
plt.rcParams['figure.figsize'] = (12, 8)
plt.rcParams['font.size'] = 11

# For inline plots
%matplotlib inline

print("✓ Libraries loaded successfully!")
# Setup: Import libraries
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
import seaborn as sns

# Set visualization style
sns.set_theme(style='whitegrid', context='notebook')
plt.rcParams['figure.figsize'] = (12, 8)
plt.rcParams['font.size'] = 11

# For inline plots
%matplotlib inline

print("✓ Libraries loaded successfully!")

Part 1: What is a Vector Field?¶

Definition¶

A vector field assigns a vector (arrow) to every point in space.

Mathematically, in 2D: $$ \mathbf{F}(x, y) = \begin{bmatrix} F_x(x, y) \\ F_y(x, y) \end{bmatrix} $$

At each point $(x, y)$, the field $\mathbf{F}$ gives us a 2D vector with:

Direction: Where the arrow points
Magnitude: How long the arrow is

Real-World Examples¶

Gravity field: Arrows point downward, magnitude depends on mass/distance
Wind field: Arrows show wind direction and speed at each location
Electric field: Arrows show force on a charged particle
Fluid flow: Arrows show velocity of water/air at each point

Visual Representation¶

We typically use:

Quiver plots: Arrows at sampled grid points
Streamlines: Curves that follow the field direction
Color maps: Color intensity shows magnitude

Let's create our first vector field!

In [ ]:

Copied!





# Example 1: Simple Linear Vector Field
# F(x, y) = (x, y) — arrows point away from origin

# Create a grid of points
x = np.linspace(-2, 2, 15)
y = np.linspace(-2, 2, 15)
X, Y = np.meshgrid(x, y)

# Define the vector field: F(x,y) = (x, y)
U = X  # x-component
V = Y  # y-component

# Compute magnitude for coloring
magnitude = np.sqrt(U**2 + V**2)

# Visualize
fig, ax = plt.subplots(figsize=(10, 10))
quiver = ax.quiver(X, Y, U, V, magnitude, cmap='viridis', alpha=0.8, scale=20)
ax.set_xlabel('x', fontsize=14)
ax.set_ylabel('y', fontsize=14)
ax.set_title('Linear Vector Field: $\\mathbf{F}(x,y) = (x, y)$', fontsize=16)
ax.grid(True, alpha=0.3)
ax.set_aspect('equal')
plt.colorbar(quiver, ax=ax, label='Magnitude')
plt.show()

print("Key Observation: Arrows point AWAY from the origin")
print("                 Magnitude increases with distance from center")
# Example 1: Simple Linear Vector Field
# F(x, y) = (x, y) — arrows point away from origin

# Create a grid of points
x = np.linspace(-2, 2, 15)
y = np.linspace(-2, 2, 15)
X, Y = np.meshgrid(x, y)

# Define the vector field: F(x,y) = (x, y)
U = X  # x-component
V = Y  # y-component

# Compute magnitude for coloring
magnitude = np.sqrt(U**2 + V**2)

# Visualize
fig, ax = plt.subplots(figsize=(10, 10))
quiver = ax.quiver(X, Y, U, V, magnitude, cmap='viridis', alpha=0.8, scale=20)
ax.set_xlabel('x', fontsize=14)
ax.set_ylabel('y', fontsize=14)
ax.set_title('Linear Vector Field: $\\mathbf{F}(x,y) = (x, y)$', fontsize=16)
ax.grid(True, alpha=0.3)
ax.set_aspect('equal')
plt.colorbar(quiver, ax=ax, label='Magnitude')
plt.show()

print("Key Observation: Arrows point AWAY from the origin")
print("                 Magnitude increases with distance from center")

Part 2: Gradient Fields — The Path to Optimization¶

What is a Gradient?¶

Given a scalar function $V(x, y)$ (called a potential), its gradient is: $$ \nabla V = \begin{bmatrix} \frac{\partial V}{\partial x} \\ \frac{\partial V}{\partial y} \end{bmatrix} $$

Key Property: The gradient points in the direction of steepest ascent!

Connection to GRL¶

In GRL, the reinforcement field $Q^+(s, a)$ is like a potential. Its gradient tells us which direction to move in state-action space to improve expected reward!

Gradient descent (going downhill) → Gradient ascent (going uphill in GRL!)

Let's visualize a gradient field.

In [ ]:

Copied!





# Example: Gradient of a Parabolic Bowl
# Potential: V(x,y) = x^2 + y^2
# Gradient: ∇V = (2x, 2y)

def potential(x, y):
    """Parabolic bowl potential"""
    return x**2 + y**2

def gradient_field(x, y):
    """Gradient of the potential"""
    return 2*x, 2*y

# Create grid (finer for smoother surface)
x = np.linspace(-2, 2, 20)
y = np.linspace(-2, 2, 20)
X, Y = np.meshgrid(x, y)

# Compute potential and gradient
Z = potential(X, Y)
U, V = gradient_field(X, Y)

# Create side-by-side visualization
fig = plt.figure(figsize=(18, 7))

# Left: Potential surface (3D)
ax1 = fig.add_subplot(121, projection='3d')
surf = ax1.plot_surface(X, Y, Z, cmap='viridis', alpha=0.8, edgecolor='none')
ax1.set_xlabel('x', fontsize=12)
ax1.set_ylabel('y', fontsize=12)
ax1.set_zlabel('$V(x,y)$', fontsize=12)
ax1.set_title('Potential Surface: $V(x,y) = x^2 + y^2$', fontsize=14)
fig.colorbar(surf, ax=ax1, shrink=0.5, label='V(x,y)')

# Right: Gradient field (2D)
ax2 = fig.add_subplot(122)
contour = ax2.contourf(X, Y, Z, levels=20, cmap='viridis', alpha=0.4)
quiver = ax2.quiver(X, Y, U, V, color='red', alpha=0.8, scale=40)
ax2.set_xlabel('x', fontsize=14)
ax2.set_ylabel('y', fontsize=14)
ax2.set_title('Gradient Field: $\\nabla V = (2x, 2y)$', fontsize=14)
ax2.set_aspect('equal')
ax2.grid(True, alpha=0.3)
fig.colorbar(contour, ax=ax2, label='V(x,y)')

plt.tight_layout()
plt.show()

print("✓ Key Insights:")
print("  1. Gradient arrows point UPHILL on the potential surface")
print("  2. At the minimum (0,0), gradient is ZERO (no arrows)")
print("  3. Magnitude grows with distance from minimum")
print("  4. To MINIMIZE: follow NEGATIVE gradient (gradient descent)")
print("  5. To MAXIMIZE: follow POSITIVE gradient (gradient ascent)")
# Example: Gradient of a Parabolic Bowl
# Potential: V(x,y) = x^2 + y^2
# Gradient: ∇V = (2x, 2y)

def potential(x, y):
    """Parabolic bowl potential"""
    return x**2 + y**2

def gradient_field(x, y):
    """Gradient of the potential"""
    return 2*x, 2*y

# Create grid (finer for smoother surface)
x = np.linspace(-2, 2, 20)
y = np.linspace(-2, 2, 20)
X, Y = np.meshgrid(x, y)

# Compute potential and gradient
Z = potential(X, Y)
U, V = gradient_field(X, Y)

# Create side-by-side visualization
fig = plt.figure(figsize=(18, 7))

# Left: Potential surface (3D)
ax1 = fig.add_subplot(121, projection='3d')
surf = ax1.plot_surface(X, Y, Z, cmap='viridis', alpha=0.8, edgecolor='none')
ax1.set_xlabel('x', fontsize=12)
ax1.set_ylabel('y', fontsize=12)
ax1.set_zlabel('$V(x,y)$', fontsize=12)
ax1.set_title('Potential Surface: $V(x,y) = x^2 + y^2$', fontsize=14)
fig.colorbar(surf, ax=ax1, shrink=0.5, label='V(x,y)')

# Right: Gradient field (2D)
ax2 = fig.add_subplot(122)
contour = ax2.contourf(X, Y, Z, levels=20, cmap='viridis', alpha=0.4)
quiver = ax2.quiver(X, Y, U, V, color='red', alpha=0.8, scale=40)
ax2.set_xlabel('x', fontsize=14)
ax2.set_ylabel('y', fontsize=14)
ax2.set_title('Gradient Field: $\\nabla V = (2x, 2y)$', fontsize=14)
ax2.set_aspect('equal')
ax2.grid(True, alpha=0.3)
fig.colorbar(contour, ax=ax2, label='V(x,y)')

plt.tight_layout()
plt.show()

print("✓ Key Insights:")
print("  1. Gradient arrows point UPHILL on the potential surface")
print("  2. At the minimum (0,0), gradient is ZERO (no arrows)")
print("  3. Magnitude grows with distance from minimum")
print("  4. To MINIMIZE: follow NEGATIVE gradient (gradient descent)")
print("  5. To MAXIMIZE: follow POSITIVE gradient (gradient ascent)")