Divergence of a Vector Field

What is the divergence of a vector field?

The divergence of a vector field A(r) is a scalar field, denoted by div A or ∇·, and is defined as the sum of the partial derivatives of its components with respect to the coordinate axes: $$ div \ A(\vec{r}) = \frac{d \ A_x (\vec{r})}{dx} + \frac{d \ A_y (\vec{r})}{dy} + \frac{d \ A_z (\vec{r})}{dz} $$

Here, r represents the position vector, which specifies the location of a point in space.

The divergence is a scalar quantity - that is, a single numerical value.

What does it tell us?

The divergence indicates the extent to which the flow lines of a vector field spread out from or converge toward a point.

It is determined by examining the behavior of the vector field’s flow.

A practical example

Consider a tank filled with water as an example of a vector field, where the vectors represent the vertical velocity of the water moving downward due to gravity.

As the tank drains, the divergence of the vector field near the outlet is negative - the water is converging toward the drain.

Farther from the outlet, the divergence is zero, as there is minimal vertical movement of the water in those regions.

And so on.