---
title: "Div and curl of grad φ, φ=log(x²+y²+z²)"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:div-curl-grad-log-x2"
source: https://www.bctengineer.com/q/semester_2/mathematics/div-curl-grad-log-x2
site: BCT Engineering
---

# Div and curl of grad φ, φ=log(x²+y²+z²)

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2071 Bhadra, 5 marks
- Video: [Find div(grad φ) and curl(grad φ) for φ = log(x² + y² + z²) | Vector Calculus](https://www.youtube.com/watch?v=1z9W-Rx9YVg) — Amat Suren Dra
- 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-log-x2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If φ = log(x²+y²+z²), find div(grad φ) and curl(grad φ).

## Short answer

φ = log(x²+y²+z²): grad φ = 2r/(x²+y²+z²); div(grad φ) = 2/(x²+y²+z²); curl(grad φ) = 0.

## Answer

φ = log(x² + y² + z²). Let s = x² + y² + z².

#### Gradient
```text
∂φ/∂x = 2x/s,   ∂φ/∂y = 2y/s,   ∂φ/∂z = 2z/s
grad φ = (2/s)(x i + y j + z k) = 2r/s
```

#### Divergence of grad φ
```text
∂/∂x (2x/s) = 2/s − 2x(2x)/s² = 2/s − 4x²/s²
```
Similarly for y and z. Add:

```text
div grad φ = 6/s − 4(x² + y² + z²)/s²
           = 6/s − 4/s = 2/s
```

**div(grad φ) = 2/(x² + y² + z²).**

#### Curl of grad φ
grad φ = (2x/s, 2y/s, 2z/s). The x-component of the curl is

```text
∂/∂y (2z/s) − ∂/∂z (2y/s)
= 2z(−2y/s²) − 2y(−2z/s²) = −4yz/s² + 4yz/s² = 0
```
The other components vanish in the same way (this is the identity curl grad φ = 0 for any twice-differentiable φ).

**curl(grad φ) = 0.**

## Check yourself

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

1. grad φ for φ = log(x²+y²+z²) is:
   A. 2r/s, s = x²+y²+z²
   B. r/s
   C. 2/s
   D. 2r s
   Answer: A. ∂φ/∂x = 2x/s. (Key point: Gradient)

2. ∂/∂x(2x/s) equals:
   A. 2/s + 4x²/s²
   B. 2/s
   C. −4x²/s²
   D. 2/s − 4x²/s²
   Answer: D. Quotient rule. (Key point: Partial derivative)

3. div grad φ equals:
   A. 6/s
   B. 0
   C. 2/(x²+y²+z²)
   D. 4/s
   Answer: C. 6/s − 4/s = 2/s. (Key point: Divergence)

4. curl grad φ equals:
   A. 2/s
   B. 0
   C. 2r/s
   D. 6/s
   Answer: B. Curl of any gradient is zero. (Key point: Curl)

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