---
title: "Div and curl of grad(x³+y³+z³−3xyz)"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:div-curl-grad-x3-y3"
source: https://www.bctengineer.com/q/semester_2/mathematics/div-curl-grad-x3-y3
site: BCT Engineering
---

# Div and curl of grad(x³+y³+z³−3xyz)

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2068 Magh, 5 marks
- Video: [Divergence and Curl of Vector Field ∇(x³ + y³ + z³ − 3xyz) | VTU Solved](https://www.youtube.com/watch?v=_wjqemMziWk) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/div-curl-grad-x3-y3
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Find the divergence and curl of F where F = grad(x³+y³+z³−3xyz).

## Short answer

F = grad(x³+y³+z³−3xyz) = (3x²−3yz, 3y²−3zx, 3z²−3xy). Divergence: div F = 6x+6y+6z = 6(x+y+z). Curl: curl F = 0, since the curl of any gradient is identically zero.

## Answer

Let φ = x³ + y³ + z³ − 3xyz and F = grad φ.

#### F
```text
∂φ/∂x = 3x² − 3yz
∂φ/∂y = 3y² − 3zx
∂φ/∂z = 3z² − 3xy
F = (3x² − 3yz) i + (3y² − 3zx) j + (3z² − 3xy) k
```

#### Divergence
```text
div F = ∂(3x² − 3yz)/∂x + ∂(3y² − 3zx)/∂y + ∂(3z² − 3xy)/∂z
      = 6x + 6y + 6z
      = 6(x + y + z)
```

#### Curl
```text
curl F = | i              j              k           |
         | ∂/∂x           ∂/∂y           ∂/∂z        |
         | 3x²−3yz        3y²−3zx        3z²−3xy     |
```
- i: ∂(3z² − 3xy)/∂y − ∂(3y² − 3zx)/∂z = −3x + 3x = 0
- j: ∂(3x² − 3yz)/∂z − ∂(3z² − 3xy)/∂x = −3y + 3y = 0
- k: ∂(3y² − 3zx)/∂x − ∂(3x² − 3yz)/∂y = −3z + 3z = 0

**Answer: div F = 6(x + y + z) and curl F = 0** (F is irrotational, as the gradient of any scalar field).

## Check yourself

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

1. ∂φ/∂x for φ = x³+y³+z³−3xyz is:
   A. 3x² + 3yz
   B. x² − yz
   C. 3x² − 3yz
   D. 3x² − 3xy
   Answer: C. Differentiate. (Key point: Gradient)

2. div F equals:
   A. 3(x + y + z)
   B. 6(x + y + z)
   C. 0
   D. 6xyz
   Answer: B. 6x + 6y + 6z. (Key point: Divergence)

3. curl F equals:
   A. 0
   B. 6(x+y+z)
   C. 3(x+y+z)
   D. i + j + k
   Answer: A. F is a gradient, hence irrotational. (Key point: Curl)

## 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

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