---
title: "Euler's theorem statement with sin u example"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:euler-s-theorem-statement-sin"
source: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-statement-sin
site: BCT Engineering
---

# Euler's theorem statement with sin u example

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2069 Poush, 5 marks; 2068 Magh, 5 marks; 2067 Chaitra, 1+4 marks (asked 3 times)
- Video: [Partial Differentiation - Euler's Theorem for Homogeneous Function](https://www.youtube.com/watch?v=bTs7ncA_AtY) — Dr.Gajendra Purohit and 2 more
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-statement-sin
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

State Euler's theorem on homogeneous functions of two independent variables. And if sin u = (√x − √y)/(√x + √y) then prove x ∂u/∂x + y ∂u/∂y = 0.

## Short answer

Euler's theorem: If f is homogeneous of degree n, x ∂f/∂x + y ∂f/∂y = n f. Let z = sin u = (√x−√y)/(√x+√y). z is homogeneous of degree 0, so x ∂z/∂x + y ∂z/∂y = 0 and substituting ∂z = cos u ∂u yields x ∂u/∂x + y ∂u/∂y = 0.

## Answer

#### Euler's theorem
If f(x, y) is a homogeneous function of degree *n* (f(tx, ty) = tⁿ f(x, y)) with continuous first partial derivatives, then

x ∂f/∂x + y ∂f/∂y = n f.

#### Proof of the given result
Given sin u = (√x − √y)/(√x + √y).

Let z = sin u = (x^(1/2) − y^(1/2))/(x^(1/2) + y^(1/2)). Replace x by tx and y by ty: the factor t^(1/2) cancels from numerator and denominator, so z(tx, ty) = t⁰ z. Thus z is homogeneous of degree n = 0.

By Euler's theorem:

x ∂z/∂x + y ∂z/∂y = 0 · z = 0

Now ∂z/∂x = cos u ∂u/∂x and ∂z/∂y = cos u ∂u/∂y, so

```text
cos u ( x ∂u/∂x + y ∂u/∂y ) = 0
x ∂u/∂x + y ∂u/∂y = 0      (cos u ≠ 0)
```

Hence proved. ∎

## Check yourself

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

1. Euler's theorem for f homogeneous of degree n states:
   A. x f_x + y f_y = n f
   B. f_x + f_y = n f
   C. x f_x + y f_y = f/n
   D. x f_y + y f_x = n f
   Answer: A. For f(tx, ty) = t^n f(x, y), x ∂f/∂x + y ∂f/∂y = n f. (Key point: Statement)

2. For z = sin u = (√x−√y)/(√x+√y), what is the degree of homogeneity?
   A. −1/2
   B. 0
   C. 1
   D. 1/2
   Answer: B. The factor t^(1/2) cancels in numerator and denominator, giving t⁰ z. (Key point: Degree)

3. Why does the proof end with x u_x + y u_y = 0?
   A. Because u is constant
   B. Euler gives 0·z = 0, and cos u(x u_x + y u_y) = 0 with cos u ≠ 0
   C. Because x = y
   D. Because sin u = 0
   Answer: B. With n = 0, x z_x + y z_y = 0; z_x = cos u u_x, so dividing by cos u gives the result. (Key point: Conclusion)

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