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University Math Lab: Systems & Structures FREE

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Harmonic Functions & Laplace's Equation

70 min University Math Lab: Systems & Structures

Learning Objectives

  • Understand harmonic functions
  • Solve Laplace's equation
  • Apply mean value property

Key Concepts

Harmonic Function

Function satisfying ∇²f = 0 (Laplace's equation)

Laplacian

∇²f = ∂²f/∂x² + ∂²f/∂y² (+ ∂²f/∂z² in 3D)

Theory

**Laplace's Equation:** ∇²f = ∂²f/∂x² + ∂²f/∂y² = 0 A function satisfying this is **harmonic**. **Properties:** - Maximum principle: A harmonic function on a closed domain attains its max/min on the boundary - Mean value property: f(center) = average of f on any surrounding circle - Real and imaginary parts of analytic functions are harmonic **Examples of harmonic functions:** - f(x,y) = x² - y² - f(x,y) = eˣcos(y) - f(x,y) = ln(x² + y²) (except at origin)

Examples

Verify f(x,y) = x² - y² is harmonic

Solution: ∂²f/∂x² = 2, ∂²f/∂y² = -2. ∇²f = 2 + (-2) = 0 ✓