Definition
In three dimensions, the problem is to find twice-differentiable real-valued functions, of real variables x, y, and z, such that
In Cartesian coordinates
In cylindrical coordinates,
In spherical coordinates,
In Curvilinear coordinates,
or
This is often written as
or, especially in more general contexts,
where ∆ = ∇² is the Laplace operator or "Laplacian"
where ∇ ⋅ = div is the divergence, and ∇ = grad is the gradient.
If the right-hand side is specified as a given function, f(x, y, z), i.e., if the whole equation is written as
then it is called "Poisson's equation".
The Laplace equation is also a special case of the Helmholtz equation.
Read more about this topic: Laplace's Equation
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