how to find laplace's equation in cgs Gaussian system
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In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as
Pierre-Simon Laplace
{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}
where {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}is the Laplace operator,[note 1] {\displaystyle \nabla \cdot }\nabla \cdot is the divergence operator (also symbolized "div"), {\displaystyle \nabla }\nabla is the gradient operator (also symbolized "grad"), and {\displaystyle f(x,y,z)}{\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function, {\displaystyle h(x,y,z)}{\displaystyle h(x,y,z)}, we have
{\displaystyle \Delta f=h.}{\displaystyle \Delta f=h.}
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.