In mathematics, Poisson's (<img class="tex" alt="\Delta\varphi=f" src="https://upload.wikimedia.org/wikipedia/en/math/5/c/1/5c1eca5688ddbfe8e4df96b205f652ed.png">)equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics.In mathematics, Laplace's(<img style="BORDER-BOTTOM-STYLE: none; LINE-HEIGHT: 19px; BORDER-LEFT-STYLE: none; FONT-FAMILY: sans-serif; BORDER-TOP-STYLE: none; BORDER-RIGHT-STYLE: none; FONT-SIZE: 13px; VERTICAL-ALIGN: middle" class="tex" alt="\nabla^2 \varphi = 0 \," src="https://upload.wikimedia.org/wikipedia/en/math/4/2/6/42635da1dea70ee9c2f855a7c52573c9.png" data-mce-style="border-style: none; vertical-align: middle; font-family: sans-serif; font-size: 13px; line-height: 19px;" data-mce-src="https://upload.wikimedia.org/wikipedia/en/math/4/2/6/42635da1dea70ee9c2f855a7c52573c9.png">) equation is a second-order partial differential equation.Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Solutions of Laplace's equation are called harmonic functions.The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.
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