- 2068 Baisakh · 4 marks
Differences between curl and divergence
Short answer
Divergence is a scalar measuring net outward flux per unit volume; curl is a vector measuring net circulation per unit area. Divergence zero means solenoidal (no sources), curl zero means irrotational (conservative). Example: ∇·D = ρ_v, ∇×E = -∂B/∂t.
What are the differences between curl and divergence?
Divergence of a vector field measures how much the field spreads out from (diverges from) a point, i.e. net outward flux per unit volume. Curl measures how much the field rotates (circulates) around a point, i.e. net circulation per unit area.
| Point | Divergence (∇ · A) | Curl (∇ × A) |
|---|---|---|
| Result | Scalar | Vector |
| Operation | Dot product of ∇ with A | Cross product of ∇ with A |
| Measures | Source or sink strength (outward flux per unit volume) | Rotation or circulation per unit area |
| Definition | lim (Δv → 0) (∮ A · dS) / Δv | lim (ΔS → 0) (∮ A · dl) / ΔS, along the normal |
| Cartesian form | ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z | determinant of [a_x a_y a_z; ∂/∂x ∂/∂y ∂/∂z; A_x A_y A_z] |
| Zero value means | Solenoidal field (no source; lines are continuous or closed) | Irrotational (conservative) field |
| Related theorem | Divergence (Gauss) theorem: ∮ A · dS = ∫ ∇ · A dv | Stokes' theorem: ∮ A · dl = ∫ (∇ × A) · dS |
| Electromagnetic example | ∇ · D = ρ_v, ∇ · B = 0 | ∇ × E = −∂B/∂t, ∇ × H = J + ∂D/∂t |
Examples:
- A = x a_x + y a_y + z a_z: ∇ · A = 3 (lines spread out from the origin), ∇ × A = 0 (no rotation).
- A = −y a_x + x a_y: ∇ · A = 0 (no spreading), ∇ × A = 2 a_z (circulates about the z-axis).
✓ Answer checked by Claude Opus 5.5 (high effort)
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