---
title: "Gradient of (a·r)/rⁿ"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:gradient-r-r"
source: https://www.bctengineer.com/q/semester_2/mathematics/gradient-r-r
site: BCT Engineering
---

# Gradient of (a·r)/rⁿ

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2073 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/gradient-r-r
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If r be the position vector and a is a constant vector then prove that ∇(a·r / rⁿ) = a/rⁿ − (n/r^(n+2)) (a·r) r.

## Short answer

Apply product rule: ∇((a·r) r^-n) = (a·r)(-n r^{-n-2} r) + r^{-n} a = a/r^n - (n/r^{n+2})(a·r) r.

## Answer

Let r = xi + yj + zk, r = |r|, and a a constant vector. We use ∇r = r/r (so ∇(rᵐ) = m r^(m−2) r) and ∇(a · r) = a.

#### Product rule
Write a · r / rⁿ = (a · r) · r⁻ⁿ and use ∇(uv) = u∇v + v∇u:

```text
∇[(a · r) r⁻ⁿ] = (a · r) ∇(r⁻ⁿ) + r⁻ⁿ ∇(a · r)
```

#### The two gradients
- ∇(r⁻ⁿ) = −n r^(−n−1) ∇r = −n r^(−n−1) (r/r) = −n r^(−n−2) r.
- ∇(a · r) = ∇(a₁x + a₂y + a₃z) = a₁i + a₂j + a₃k = a.

#### Combine
```text
∇(a·r / rⁿ) = (a · r)(−n r^(−n−2) r) + r⁻ⁿ a
            = a/rⁿ − (n / r^(n+2)) (a · r) r
```

**Hence ∇(a·r / rⁿ) = a/rⁿ − (n/r^(n+2)) (a · r) r.**

## Check yourself

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

1. ∇(uv) equals:
   A. u∇v − v∇u
   B. ∇u ∇v
   C. u v ∇
   D. u∇v + v∇u
   Answer: D. Applied with u = a·r and v = r^(−n). (Key point: Product rule)

2. ∇(r^(−n)) equals:
   A. −n r^(−n−1)
   B. n r^(−n−2) r
   C. −n r^(−n−2) r
   D. −n r^(−n) r
   Answer: C. ∇r = r/r. (Key point: Gradient of r^(−n))

3. ∇(a · r) is:
   A. r
   B. a
   C. a·r
   D. 0
   Answer: B. Differentiate a1x + a2y + a3z. (Key point: Gradient of a·r)

4. The final result is:
   A. a/rⁿ − n(a·r) r/r^(n+2)
   B. a/rⁿ + n(a·r) r/r^(n+2)
   C. n a/rⁿ
   D. a/r^(n+2)
   Answer: A. Combine the two terms. (Key point: Result)

## Close variants set in other exams

- [Prove (a×∇)×r = −2a](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aprove-r-2a) — Mathematics, chapter 5, asked 6×
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