---
title: "Differences between curl and divergence"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:curl-vs-divergence-differences"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/curl-vs-divergence-differences
site: BCT Engineering
---

# Differences between curl and divergence

Electromagnetism · Chapter 1: Introduction

- Asked in: 2068 Baisakh, 4 marks
- Video: [Divergence and curl:  The language of Maxwell's equations, fluid flow, and more](https://www.youtube.com/watch?v=rB83DpBJQsE) — 3Blue1Brown
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/curl-vs-divergence-differences
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

What are the 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.

## Answer

**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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the nature of divergence and curl results?
   A. Both are scalars
   B. Divergence is a scalar, curl is a vector
   C. Divergence is a vector, curl is a scalar
   D. Both are vectors
   Answer: B. Divergence uses the dot product so it is a scalar; curl uses the cross product so it is a vector. (Key point: Scalar vs vector)

2. What does each quantity measure?
   A. Divergence: rotation; curl: source strength
   B. Divergence: net outward flux per unit volume; curl: circulation per unit area
   C. Both measure flux per volume
   D. Both measure circulation
   Answer: B. Divergence shows source or sink strength; curl shows rotation of the field. (Key point: What each measures)

3. What does a zero value mean for each?
   A. Div = 0: irrotational; curl = 0: solenoidal
   B. Div = 0: solenoidal; curl = 0: irrotational (conservative)
   C. Both mean the field is zero
   D. Both mean conservative
   Answer: B. Zero divergence means solenoidal (no sources) and zero curl means irrotational (conservative). (Key point: Zero values)

4. For A = x a_x + y a_y + z a_z, what are div and curl?
   A. div = 0, curl = 0
   B. div = 3, curl = 0
   C. div = 0, curl = 3
   D. div = 3, curl = 3
   Answer: B. ∇·A = 1 + 1 + 1 = 3 (lines spread from the origin) and the curl is zero (no rotation). (Key point: Example field)

## Related in other subjects

- [Divergence and curl identity](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adivergence-and-curl-identity) — Mathematics III, chapter 2
- [Div and curl of grad(x³+y³+z³−3xyz)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adiv-curl-grad-x3-y3) — Mathematics, chapter 5