Laplace's equation is a second-order partial differential equation that describes the behavior of scalar fields, such as temperature or electric potential, in a given region. It is expressed as ∇²φ = 0, where φ represents the scalar field and ∇² is the Laplacian operator. This equation is fundamental in various fields, including physics, engineering, and mathematics. Solutions to Laplace's equation are called harmonic functions, which have the property of being smooth and continuous. These functions arise in many physical situations, such as steady-state heat conduction and electrostatics, where there are no sources or sinks within the region of interest.