Euler’s Theorem – Differential Calculus for GATE, ESE, AE/JE

Euler’s Theorem is an important result in Differential Calculus related to homogeneous functions. It is frequently asked in GATE, ESE, AE/JE exams and helps simplify partial derivative problems.

Homogeneous Function

A function f(x, y) is said to be homogeneous of degree n if:

f(λx, λy) = λⁿ f(x, y)

Euler’s Theorem Statement

If f(x, y) is homogeneous of degree n, then:

x (∂f/∂x) + y (∂f/∂y) = n f(x, y)

Extension to Three Variables

If f(x, y, z) is homogeneous of degree n:

x (∂f/∂x) + y (∂f/∂y) + z (∂f/∂z) = n f(x, y, z)

Steps to Apply Euler’s Theorem

1. Check if function is homogeneous
2. Find degree (n)
3. Apply formula directly

Solved Examples

Example 1

f(x, y) = x² + y²

Degree = 2

∂f/∂x = 2x, ∂f/∂y = 2y

x(2x) + y(2y) = 2(x² + y²)

Verified

Example 2

f(x, y) = x³ + 3xy²

Degree = 3

∂f/∂x = 3x² + 3y²
∂f/∂y = 6xy

x(3x² + 3y²) + y(6xy)
= 3x³ + 3xy² + 6xy²
= 3(x³ + 3xy²)

Verified

Example 3

If f(x, y) is homogeneous of degree 2, then:

x (∂f/∂x) + y (∂f/∂y) = 2f

Important Results

• Works only for homogeneous functions
• Saves time in partial derivative problems
• Very common in conceptual questions

Shortcut Tricks

• Check degree quickly (sum of powers)
• Apply formula directly without full differentiation (in MCQs)
• Use in reverse problems (finding degree)

Questions

Q1. Euler’s theorem is applicable to?

Answer: Homogeneous functions

Q2. If degree = n, result equals?

Answer: n f(x,y)

Exam Tips

• Identify homogeneous functions quickly
• Memorize formula properly
• Avoid unnecessary differentiation
• Practice PYQs

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