What is the physical meaning of the divergence of a vector field at a specific point?

Conditions

• Applicable to any continuous vector field
• Evaluated at a specific point $M$

Reasoning, step by step

1. Imagine shrinking a control volume around point $M$ to an infinitesimal size.
2. Observe the net flow crossing the boundary of this tiny volume.
3. If more flows out than in, the divergence is positive (source).
4. If more flows in than out, the divergence is negative (sink).
5. If flow in equals flow out, the divergence is zero.

Example

Text at the bottom summarizes: 'Divergence represents the intensity of dispersion of the vector field at point M'.

Common misconceptions

• Confusing divergence with curl; divergence measures spreading, while curl measures rotation.
• Assuming divergence implies creation of matter; in physics, it often relates to sources/sinks in fields like fluid dynamics or electromagnetism.

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