What are two and three dimensional Laplace equations?

Welcome back MechanicaLEi, did you know that two dimensional Laplace equation is used to validate an electrostatic potential and that three-dimensional Laplace equation helps solve the Schrödinger’s equation? This makes us wonder, what are two and three dimensional Laplace equations?
Before we jump in check out the previous part of this series to learn about what applications of Partial differential equations are?
Now, The Laplace equation in two independent variables has the form dou squared psi upon dou x squared plus dou squared psi upon dou y squared represented as psi xx plus psi yy equals to zero. The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z equals to x plus i y, and if f of z equals to u of x comma y plus i v of x comma y, then the necessary condition that f of z be analytic is that the Cauchy–Riemann equations be satisfied that is, u x equals to v y and u x equals to minus u y. Where ux is the first partial derivative of u with respect to x. It follows that u yy equals to minus v x of y equals to minus v y of x equals to minus u x of x. Therefore, u and v similarly satisfy the Laplace equation.
In three-dimensions, Laplace's equation is given by dou squared psi upon dou x squared plus dou squared psi upon dou y squared plus dou squared psi upon dou z squared equals to zero. We look for solutions that are expressible as products of a function of x alone, X of x, a function of y alone, Y of y, and a function of z alone, Z of z. Introducing the second equation into the first and dividing by psi, we obtain one upon x into d squared x upon d x squared plus one upon y into d squared Y upon d y squared plus one upon z into d squared z upon d z squared equals to zero. Hence, we first saw what Two dimensional Laplace equations are and then went on to see what three dimensional Laplace equations are?
In the next episode of MechanicaLEi find out what Solid Geometry is?

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