Introduction to Vector Fields

On a spring day over the San Francisco Bay, the wind can move differently at each location. To visualize this movement on a two-dimensional map, an arrow is placed at coordinates across the region.

This physical phenomenon is described mathematically as a vector field, defined by the function F(x, y). To visualize this field, a map of assigned vectors is drawn, showing the field's direction and magnitude at different locations.

The behavior of a vector field can often be determined from its component-wise definition.

These components define the flow. For example, if the x-component is zero and the y-component is constant, every arrow is identical—showing wind moving at the same speed and direction everywhere.

But if the y-component increases as x increases, the arrows grow longer, indicating that the field increases in magnitude as one moves across the map. If the y-component decreases as x increases, the field's magnitude gradually dies out.

Visualization often helps interpret a vector field directly. Plotting these arrows reveals how wind patterns can vary across a bay, with higher magnitudes over open water and steady swirls near the shore.