A scalar field gives you a number at every point. A vector field gives you an arrow — a direction and a magnitude — at every point.

Examples

• Wind: At every point in the atmosphere, the wind has a speed and a direction. That’s a vector field.
• Ocean currents: At every point in the ocean, water flows in some direction at some speed.
• Gravity: At every point near the Earth, there’s a gravitational pull pointing downward with a specific strength.
• Electric fields: At every point around a charged particle, there’s a force that would push or pull another charge.

In each case: pick a point → get an arrow.

Mathematical notation

A 2D vector field is a function $\mathbf{F}(x, y)$ that returns a vector at each point:

$$\mathbf{F}(x, y) = \begin{pmatrix} F_x(x, y) \\ F_y(x, y) \end{pmatrix}$$

The two components $F_x$ and $F_y$ give the arrow’s horizontal and vertical parts.

For example, the vortex field $\mathbf{F}(x, y) = (-y, x)$ points perpendicular to the line from the origin — everything swirls.

Two key questions

Once you have a vector field, two natural questions arise:

1. Does the field have sources or sinks? Is stuff being created or destroyed anywhere? This leads to divergence.
2. Does the field rotate? Is there swirling motion? This leads to curl.

These two operations — divergence and curl — are the mathematical heart of Maxwell’s equations.