What is the rigorous limit definition of the divergence of a vector field at a point M?

Conditions

• The volume $V$ shrinks to the point $M$ (radius tends to zero)
• The vector field $\vec{F}$ is smooth in the neighborhood of $M$

Reasoning, step by step

1. Consider a small closed surface $S$ surrounding point $M$ with enclosed volume $V$.
2. Calculate the flux $\iint_S \vec{F} \cdot \vec{n} dS$.
3. Normalize by dividing by volume $V$.
4. Take the limit as $V \to 0$.
5. The resulting value is the divergence at $M$.

Example

The core defining equation appears as: $\lim_{V \to 0} \frac{1}{V} \iint_S \vec{F} \cdot \vec{n} dS = \text{div } \vec{F}(M)$.

Common misconceptions

• Assuming the shape of the shrinking volume matters; for smooth fields, the limit is independent of shape.
• Thinking divergence is only about the boundary; it is a local property derived from the infinitesimal limit.

Watch the explanation

Explore next

• What is the physical interpretation of a positive divergence value?
• How does this relate to the partial derivative sum form shown earlier?