Curl and Divergence of Vector Fields

Imagine leaves floating on a river. Some areas swirl like small whirlpools, while other areas spread apart. These two distinct motions are the fundamental behaviors of any vector field.

These two behaviors are known as curl and divergence, respectively.

Curl measures how much a vector field rotates around a point. Mathematically, it is calculated by taking the cross product of the del operator with the vector field.

On the other hand, divergence measures the net flow at a point. It is calculated as the dot product of the del operator with the vector field.

In a vector field that spreads out to the right, divergence is positive. Since there is no rotation, the curl is zero.

Now consider a circular vector field where vectors loop around a central point. This field has a nonzero curl, but the divergence is zero because the rotation doesn’t introduce any spreading.

These operators are fundamental to the laws of electromagnetism. Gauss’s law uses divergence to find electric charge density, while Faraday’s law uses curl to show how a changing magnetic field creates an electric field.