Euler’s theorem is an important result in differential calculus. It is used for functions that depend on two or more variables. The theorem connects the degree of a homogeneous function with its partial derivatives.
Homogeneous Functions
A function f(x, y) is called homogeneous of degree n if:
f(tx, ty) = tⁿ · f(x, y)
This means multiplying both variables by a constant t multiplies the entire function by tⁿ.
Examples of Homogeneous Functions
- f(x, y) = x² + y² → degree 2
- f(x, y) = x³ + 3x²y → degree 3
- f(x, y) = x²y → degree 3
Euler’s Theorem
If f(x, y) is homogeneous of degree n, then:
x (∂f/∂x) + y (∂f/∂y) = n f(x, y)
Meaning of the Theorem
The theorem states that for a homogeneous function, a weighted sum of the first-order partial derivatives gives back the function multiplied by its degree n.
Proof Idea (Simple Explanation)
Start from the definition:
f(tx, ty) = tⁿ f(x, y)
Differentiate both sides with respect to t, then put t = 1. That leads directly to the Euler relation:
x fx + y fy = n f
Extended Euler’s Theorem (Higher-Order Form)
If f(x, y) is homogeneous of degree n, then:
x² fxx + 2xy fxy + y² fyy = n(n − 1) f(x, y)
Usage
Used especially in optimisation, economics, engineering and chemical kinetics.
Partial Derivatives of Homogeneous Functions
If f is homogeneous of degree n:
- ∂f/∂x is homogeneous of degree (n − 1)
- ∂f/∂y is homogeneous of degree (n − 1)
Applications of Euler’s Theorem
- Simplifying functions before differentiation
- Checking whether a function is homogeneous
- Used in thermodynamics, economics, fluid mechanics
- Used to find unknown functions using degree and derivative relations
Examples (Conceptual Overview)
Example 1: Verify Euler’s theorem for f(x, y) = x² + y²
Degree n = 2
Compute partials:
- fx = 2x
- fy = 2y
Check:
x(2x) + y(2y) = 2(x² + y²)
The theorem is verified.
Example 2: f(x, y) = x³ + 3x²y
Degree = 3 Apply Euler relation to verify.
Practice Questions
- Check if f(x, y) = x³ + y³ is homogeneous. If yes, find its degree.
- Verify Euler’s theorem for f(x, y) = x²y.
- If f(x, y) is homogeneous of degree 4, show that x fx + y fy = 4f.
- Use extended Euler theorem for f(x, y) = x²y to compute second derivatives.
Detailed Notes:
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