Curl and divergence are two ways to measure the local behavior of a vector field. A vector field assigns a vector to every point in space, such as wind velocity in the atmosphere or electric field around charges. Curl tells whether the field tends to make tiny objects rotate, while divergence tells whether the field tends to spread out from or flow into a point.
These ideas matter because they turn complex field patterns into precise mathematical quantities.
In three dimensions, curl is a vector that points along the axis of local rotation, and its magnitude measures the strength of that rotation. Divergence is a scalar that measures net outward flow per unit volume near a point. In physics, curl appears in rotating fluids, magnetic fields, and electromagnetic induction, while divergence appears in fluid sources, electric charge, and conservation laws.
Together, they help describe how fields circulate and how they expand or compress.
Key Facts
- For a vector field F = <P, Q, R>, divergence is div F = ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
- For a vector field F = <P, Q, R>, curl is curl F = ∇ × F = <∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y>.
- Positive divergence means the field has net outward flow near a point, like a source.
- Negative divergence means the field has net inward flow near a point, like a sink.
- In two dimensions, for F = <P, Q>, the scalar curl component is ∂Q/∂x - ∂P/∂y.
- A field can have zero divergence but nonzero curl, or zero curl but nonzero divergence, so the two measurements describe different behaviors.
Vocabulary
- Vector field
- A vector field assigns a vector with magnitude and direction to each point in a region of space.
- Divergence
- Divergence is a scalar measure of how much a vector field flows outward from or inward toward a point.
- Curl
- Curl is a vector measure of the local rotation or circulation tendency of a vector field.
- Del operator
- The del operator ∇ is a vector differential operator used to write gradient, divergence, and curl formulas compactly.
- Source
- A source is a location where field lines or flow appear to spread outward, giving positive divergence.
Common Mistakes to Avoid
- Confusing curl with divergence is wrong because curl measures local rotation, while divergence measures net outward or inward flow.
- Treating divergence as a vector is wrong because divergence produces a scalar value at each point, not a direction in space.
- Forgetting the order of terms in curl is wrong because changing the order can reverse signs and give the wrong rotation direction.
- Assuming zero divergence means the field is zero is wrong because a field can circulate strongly while having no net source or sink.
Practice Questions
- 1 For F = <2x, 3y, -z>, compute div F at any point.
- 2 For the two-dimensional field F = <-y, x>, compute the scalar curl ∂Q/∂x - ∂P/∂y and the divergence ∂P/∂x + ∂Q/∂y.
- 3 A small paddle wheel placed in a flowing liquid spins in place, but dye near the same point does not spread outward or collect inward. What does this suggest about the curl and divergence near that point?