Laplace Equation Derivation

PDEs · Axiom Academy

LESSON Laplace Equation Derivation Deriving Laplace's Equation from Heat Equation, Electrostatics, and Fluid Flow 1. Derivation from Heat Equation The heat equation describes how temperature evolves over time in a conducting medium. Let's see what happens when the temperature reaches equilibrium. At Steady State: When the system reaches thermal equilibrium, the temperature no longer changes with time. This means the time derivative vanishes: Since k > 0, we can divide both sides by k to obtain: 2. Derivation from Electrostatics In electrostatics, the electric potential φ determines the electric field. Let's derive Laplace's equation from fundamental electromagnetic laws. The electric field is the negative gradient of the potential Gauss's law relates the divergence of E to the charge density Combining the equations: Substitute the expression for E into Gauss's law: This is Poisson's equation . In charge-free regions (ρ = 0), it reduces to: 3. Derivation from Incompressible Fluid Flow For incompressible, irrotational fluid flow, we can introduce a velocity potential. Let's see how this leads to Laplace's equation. Incompressible: The fluid density is constant (∇·v = 0) Irrotational: The fluid has no vorticity (∇×v = 0) Velocity Potential: For irrotational flow, we can define a velocity potential φ such that: Substituting into incompressibility: Replace v with ∇φ in the incompressibility condition: 4. General Form in Different Coordinate Systems

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