 Welcome back MechanicalEI. Did you know that 2-dimensional Laplace equation is used to validate an electrostatic potential and 3-dimensional Laplace equation helps solve the Schrodinger's equation? This makes us wonder what are 2 and 3-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 0. The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z equals to x plus iy and if f of z equals to u of x comma y plus i into 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, ux equals to vy and ux equals to minus uy, where ux is the first partial derivative of u with respect to x. It follows that uyy equals to minus vx of y equals to minus vy of x equals to minus ux of x. Therefore, u and v similarly satisfy the Laplace equation. In three dimensions, Laplace 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 0. 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 1 upon x into d squared x upon dx squared plus 1 upon y into d squared y upon dy squared plus 1 upon z into d squared z upon dz squared equals to 0. Hence, we first saw what two-dimensional Laplace equations are, and then went on to see what three-dimensional Laplace equations are. So, here are the top mechanical EIs of our last videos. In the next episode of Mechanical EI, find out what solid geometry is.