---
title: "Divergence and curl identity"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:divergence-and-curl-identity"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/divergence-and-curl-identity
site: BCT Engineering
---

# Divergence and curl identity

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2067 Magh
- Video: [Vector calculus identities | Lecture 21 | Vector Calculus for Engineers](https://www.youtube.com/watch?v=PCIQueiemzY) — Jeffrey Chasnov
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/divergence-and-curl-identity
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Define divergence and curl of V. Prove that div(Curl V) = 0.

## Short answer

Divergence: ∇·V = ∂V₁/∂x + ∂V₂/∂y + ∂V₃/∂z. Curl: ∇×V = (∂V₃/∂y−∂V₂/∂z)i + (∂V₁/∂z−∂V₃/∂x)j + (∂V₂/∂x−∂V₁/∂y)k. Then div(curl V) = ∂²V₃/∂x∂y − ∂²V₂/∂x∂z + … = 0 because mixed partial derivatives are equal.

## Answer

#### Divergence
For a vector function V = V₁ i + V₂ j + V₃ k, the divergence is the scalar

div V = ∇·V = ∂V₁/∂x + ∂V₂/∂y + ∂V₃/∂z.

It measures the net outflow of the field per unit volume at a point.

#### Curl
```text
curl V = ∇ × V = | i    j    k  |
                 | ∂x   ∂y   ∂z |
                 | V₁   V₂   V₃ |
       = (∂V₃/∂y − ∂V₂/∂z) i + (∂V₁/∂z − ∂V₃/∂x) j + (∂V₂/∂x − ∂V₁/∂y) k
```

It measures the rotation of the field at a point.

#### Proof that div(curl V) = 0
Let W = curl V, with components W₁ = ∂V₃/∂y − ∂V₂/∂z, W₂ = ∂V₁/∂z − ∂V₃/∂x, W₃ = ∂V₂/∂x − ∂V₁/∂y.

```text
div W = ∂W₁/∂x + ∂W₂/∂y + ∂W₃/∂z
      = ∂²V₃/∂x∂y − ∂²V₂/∂x∂z
      + ∂²V₁/∂y∂z − ∂²V₃/∂y∂x
      + ∂²V₂/∂z∂x − ∂²V₁/∂z∂y
```

If V has continuous second partial derivatives, the mixed partials are equal (∂²V₃/∂x∂y = ∂²V₃/∂y∂x, etc.). Each pair cancels:

```text
(V₃: xy − yx) + (V₂: zx − xz) + (V₁: yz − zy) = 0
```

Hence **div(curl V) = ∇·(∇ × V) = 0**. (Also, ∇·(∇ × V) is the scalar triple product [∇, ∇, V], which vanishes because two rows are identical.)

## Check yourself

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

1. Divergence of V is:
   A. A vector with components of partial derivatives
   B. The determinant with i, j, k
   C. The scalar dV1/dx + dV2/dy + dV3/dz
   D. dV1/dy + dV2/dz + dV3/dx
   Answer: C. div V = nabla.V, a scalar measuring net outflow. (Key point: Divergence)

2. Curl V is:
   A. A scalar measuring outflow
   B. The sum of second derivatives
   C. The gradient of V
   D. A vector measuring rotation, given by the determinant nabla x V
   Answer: D. Curl is a vector computed from the determinant. (Key point: Curl)

3. Why does div(curl V) = 0?
   A. Mixed partial derivatives are equal and cancel in pairs
   B. Curl V is always a constant
   C. V is assumed zero
   D. The divergence of any vector is zero
   Answer: A. Each mixed partial appears once with + and once with - sign. (Key point: Why zero)

4. What condition is needed in the proof?
   A. V must be constant
   B. Continuous second partial derivatives of V
   C. V must be conservative
   D. Only first partials exist
   Answer: B. This ensures equality of mixed partials. (Key point: Condition)

## Related in Mathematics III

- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-three) — Mathematics III, chapter 1, asked 9×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-five) — Mathematics III, chapter 1, asked 9×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-13) — Mathematics III, chapter 1, asked 9×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-14) — Mathematics III, chapter 1, asked 9×
- [Determinant identity: −(b − a)⁴](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-b-a) — Mathematics III, chapter 1, asked 7×
- [Determinant identity: 4a²b²c²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-4a2b2c2) — Mathematics III, chapter 1, asked 6×

## Related in other subjects

- [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
- [Differences between curl and divergence](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acurl-vs-divergence-differences) — Electromagnetism, chapter 1
- [Curl of a×b](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acurl-b) — Mathematics, chapter 5, asked 2×