---
title: "Evaluate using Stokes' theorem"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-stokes-theorem-five"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-stokes-theorem-five
site: BCT Engineering
---

# Evaluate using Stokes' theorem

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

- Asked in: 2072 Kartik, 5 marks
- Video: [Stoke's Theorem Problem 1 |  Vector Claculus | Engineering Mathematics](https://www.youtube.com/watch?v=DRWfHYiXnBs) — Alex Maths Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-stokes-theorem-five
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Apply Stokes' theorem to evaluate ∮_C (x + y)dx + (2x − z)dy + (y + z)dz where C is the boundary of the triangle with vertices (2,0,0), (0,3,0) and (0,0,6).

## Short answer

curl F = 2i + k. Triangle lies on plane 3x+2y+z=6. (curl F)·n dS = 7 dx dy. Area of projection = 3, so integral = 21.

## Answer

**Stokes' theorem:** ∮_C F·dr = ∬_S (curl F)·n dS, with F = (x + y) i + (2x − z) j + (y + z) k.

The triangle with vertices A(2,0,0), B(0,3,0), C(0,0,6) lies on the plane x/2 + y/3 + z/6 = 1, that is 3x + 2y + z = 6.

```text
curl F = (∂R/∂y − ∂Q/∂z) i + (∂P/∂z − ∂R/∂x) j + (∂Q/∂x − ∂P/∂y) k
       = (1 + 1) i + (0 − 0) j + (2 − 1) k = 2 i + k

n = (3i + 2j + k)/√14        (upward, for A → B → C anticlockwise seen from above)
(curl F)·n = (6 + 0 + 1)/√14 = 7/√14
dS = dx dy/|n·k| = √14 dx dy
```

So (curl F)·n dS = 7 dx dy. The projection on the xy-plane is the triangle with vertices (0,0), (2,0), (0,3), of area ½ · 2 · 3 = 3.

```text
∮_C F·dr = 7 × 3 = 21
```

**Check (direct):** summing the line integrals along A→B, B→C, C→A gives 21 as well.

**Answer: 21** (for the orientation A → B → C → A).

## Check yourself

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

1. Plane through (2,0,0),(0,3,0),(0,0,6):
   A. x+y+z=6
   B. 2x+3y+z=6
   C. 3x + 2y + z = 6
   D. 3x+2y+z=1
   Answer: C. Intercepts 2, 3, 6. (Key point: Plane)

2. curl F is:
   A. i + k
   B. 2 i
   C. 2 i + j + k
   D. 2 i + k
   Answer: D. (1+1)i + 0 j + (2-1)k. (Key point: Curl)

3. (curl F).n dS =
   A. 7 dx dy
   B. 7/sqrt14 dx dy
   C. 2 dx dy
   D. sqrt14 dx dy
   Answer: A. 7/sqrt14 times sqrt14. (Key point: Integrand)

4. Value:
   A. 7
   B. 21
   C. 14
   D. 42
   Answer: B. 7 times projected area 3. (Key point: Result)

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