Divergence and Curl of a Vector Field: Concepts and Examples

A vector field assigns an arrow to every point in space — think of wind velocity across a weather map or water flow in a river. Two questions matter most about such a field: at a given point, is more fluid flowing out than in (is it a source or a sink)? And is the fluid swirling around that point? Divergence answers the first question; curl answers the second.

For a vector field F = (P, Q, R), the divergence div F = ∇·F = Px + Qy + Rz is a scalar measuring net outflow per unit volume. The curl curl F = ∇×F is a vector measuring rotation: its direction is the axis of swirl (right-hand rule) and its length is twice the angular speed. Together they describe the local “personality” of any field, from electromagnetism to fluid flow.

Key Concepts

• Divergence is a scalar: positive = source (fluid created), negative = sink, zero = incompressible (what flows in flows out).
• Curl is a vector: zero = irrotational (no local spinning); a nonzero curl’s direction is the rotation axis.
• Del products: ∇·F uses the dot product of del with F; ∇×F uses the cross product, computed with the determinant formula.
• If F = ∇f (a gradient field), then curl F = 0 always.
• Units: if F is a velocity in m/s, both div F and curl F have units 1/s.

Formulas

• div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
• curl F = (Ry − Qz, Pz − Rx, Qy − Px)
• As a determinant: |i  j  k; ∂/∂x  ∂/∂y  ∂/∂z; P  Q  R|

Worked Example

For F(x, y, z) = (x²y, xyz, yz²), compute div F and curl F at the point (1, 2, 1).

1. Divergence. Px = 2xy, Qy = xz, Rz = 2yz, so div F = 2xy + xz + 2yz.
2. Evaluate: div F(1, 2, 1) = 2(1)(2) + (1)(1) + 2(2)(1) = 4 + 1 + 4 = 9 per second — net outflow, so the point behaves like a source.
3. Curl components. curl F = (Ry − Qz, Pz − Rx, Qy − Px) = (z² − xy, 0 − 0, yz − x²).
4. Evaluate: curl F(1, 2, 1) = (1 − 2, 0, 2 − 1) = (−1, 0, 1).
5. Interpretation: at (1, 2, 1) the field is expanding (divergence 9 > 0) and swirling about an axis pointing along (−1, 0, 1) — a tiny paddle wheel placed there would spin around that axis.

Common Misconceptions

• “Divergence measures how fast the field moves.” Speed is |F|; divergence measures net outflow, which is zero even in a fast uniform flow.
• “Curl is a number.” Curl is a vector — its direction (the rotation axis) matters as much as its magnitude.
• “Zero curl means the field is zero.” A uniform flowing river has zero curl everywhere but nonzero velocity; zero curl means no local rotation, nothing more.

Real-World Applications

• Electromagnetism: Maxwell’s equations are written with divergence (Gauss’s laws) and curl (Faraday’s and Ampère’s laws).
• Fluid dynamics: divergence locates sources and sinks; curl (vorticity) describes whirlpools and turbulence.
• Weather: divergence in wind fields signals rising or sinking air.
• Computer graphics: curl-based noise creates natural-looking swirling smoke and fire.

Divergence and curl are the two local diagnostics of a vector field: divergence asks “how much flows out?” and curl asks “how much spins?” Memorize div F = Px + Qy + Rz and the cyclic pattern of the curl components, and you can read the behavior of any field — wind, water, or electromagnetic — point by point.