---
title: "Curl of a×b"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:curl-b"
source: https://www.bctengineer.com/q/semester_2/mathematics/curl-b
site: BCT Engineering
---

# Curl of a×b

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2068 Bhadra, 5 marks; 2068 Magh, 5 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/curl-b
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Prove that curl(a×b) = a div b − (a·∇)b. [Right side as printed; b after (a·∇) is probably intended]

## Short answer

For constant a, expand curl(a×b) componentwise, add and subtract a₁∂b₁/∂x, to obtain a div b − (a·∇)b. Hence curl(a×b) = a div b − (a·∇)b.

## Answer

[Reading: a is a constant vector (this is the case in which the printed result is true), and the term (a · ∇)b means the operator a₁∂/∂x + a₂∂/∂y + a₃∂/∂z applied to b. For a variable a the full identity is curl(a×b) = a div b − b div a + (b·∇)a − (a·∇)b, which reduces to the printed one when a is constant.]

Let a = (a₁, a₂, a₃) be constant and b = (b₁, b₂, b₃), functions of x, y, z.

a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k.

#### x-component of the curl
```text
[curl(a×b)]ₓ = ∂/∂y (a₁b₂ − a₂b₁) − ∂/∂z (a₃b₁ − a₁b₃)
             = a₁ ∂b₂/∂y − a₂ ∂b₁/∂y − a₃ ∂b₁/∂z + a₁ ∂b₃/∂z
```
Add and subtract a₁ ∂b₁/∂x:

```text
= a₁ (∂b₁/∂x + ∂b₂/∂y + ∂b₃/∂z) − (a₁ ∂/∂x + a₂ ∂/∂y + a₃ ∂/∂z) b₁
= a₁ (div b) − (a · ∇) b₁
```

#### Other components
The y and z components follow by cyclic change of (x, y, z) and (1, 2, 3):

```text
[curl(a×b)]_y = a₂ div b − (a·∇)b₂
[curl(a×b)]_z = a₃ div b − (a·∇)b₃
```

**Hence curl(a × b) = a div b − (a · ∇)b**, for constant a.

## Check yourself

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

1. The identity as proved holds for:
   A. any a
   B. constant b only
   C. constant a
   D. solenoidal b
   Answer: C. For variable a more terms appear. (Key point: Assumption)

2. To get a1 div b, the proof:
   A. integrates by parts
   B. adds and subtracts a1 ∂b1/∂x
   C. uses ∇·a = 0
   D. sets b1 = 0
   Answer: B. This completes div b in the x-component. (Key point: Trick)

3. (a · ∇) is:
   A. a1 ∂/∂x + a2 ∂/∂y + a3 ∂/∂z
   B. a1 ∂/∂x × a2 ∂/∂y
   C. a × ∇
   D. ∇ · a
   Answer: A. A scalar differential operator applied to b. (Key point: Meaning of operator)

4. How are the y and z components obtained?
   A. By setting a2 = a3 = 0
   B. By integrating
   C. They are zero
   D. By cyclic change of (x,y,z) and (1,2,3)
   Answer: D. Symmetry gives a2 div b − (a·∇)b2 and so on. (Key point: Components)

## Related in Mathematics

- [Div and curl of grad φ, φ=log(x²+y²+z²)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adiv-curl-grad-log-x2) — Mathematics, chapter 5
- [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

## Related in other subjects

- [Setting out right hand side composite curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aright-hand-composite-curve-setting-out) — Surveying II, chapter 6
- [Divergence and curl identity](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adivergence-and-curl-identity) — Mathematics III, chapter 2
- [Flow lines cut equipotentials at right angles](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aprove-flow-equipotential-orthogonal) — Soil Mechanics, chapter 7, asked 3×
- [H due to side of triangular loop](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Ah-due-to-triangle-side-1) — Electromagnetism, chapter 3