---
title: "Stream function and complex potential"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:stream-function-and-complex-potential"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/stream-function-and-complex-potential
site: BCT Engineering
---

# Stream function and complex potential

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

- Asked in: 2082 Kartik (new course), 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/stream-function-and-complex-potential
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

An electrostatic field in a plane is given by the potential function φ(x, y) = 3x²y − y³. Find the corresponding stream function and complex potential f(z).

## Short answer

The stream function is ψ = 3xy^2 - x^3 + c. The complex potential is f(z) = φ + iψ = -i z^3 + c'.

## Answer

The complex potential is f(z) = φ + iψ, where f is analytic and φ, ψ satisfy the Cauchy–Riemann equations φ_x = ψ_y, φ_y = −ψ_x.

#### Check that φ is harmonic
```text
φ = 3x²y − y³
φ_xx = 6y,  φ_yy = −6y  → φ_xx + φ_yy = 0
```

#### Stream function
```text
ψ_y = φ_x = 6xy   → ψ = 3xy² + g(x)
ψ_x = 3y² + g'(x) = −φ_y = −(3x² − 3y²) = 3y² − 3x²
g'(x) = −3x²  → g = −x³
ψ = 3xy² − x³ + c
```

#### Complex potential
Since z³ = (x³ − 3xy²) + i(3x²y − y³), we have φ = Im(z³) and ψ = −Re(z³):

```text
f(z) = φ + iψ = (3x²y − y³) + i(3xy² − x³) = −i z³ + c
```

Check: −i z³ = −i(x³ − 3xy²) + (3x²y − y³) ✓.

**Answer: ψ = 3xy² − x³ + c and f(z) = −i z³ + c'.**

## Check yourself

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

1. The complex potential is:
   A. f = phi - psi
   B. f = phi psi
   C. f = phi + i psi, analytic
   D. f = phi_x + i phi_y
   Answer: C. phi and psi satisfy Cauchy-Riemann. (Key point: Complex potential)

2. Why verify phi_xx + phi_yy = 0?
   A. to find psi directly
   B. to find the constant
   C. to check continuity
   D. phi must be harmonic to be the real/imag part of an analytic function
   Answer: D. Only harmonic phi has a conjugate. (Key point: Harmonic check)

3. psi is:
   A. 3xy^2 - x^3 + c
   B. x^3 - 3xy^2 + c
   C. 3x^2y - y^3
   D. 3xy^2 + x^3
   Answer: A. From psi_y = 6xy and psi_x = -phi_y. (Key point: Stream function)

4. f(z) is:
   A. i z^3
   B. -i z^3 + c
   C. z^3
   D. -z^3
   Answer: B. phi = Im(z^3), psi = -Re(z^3). (Key point: f(z))

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