Welcome back Mechanical EI. Did you know that two dimensional laplace equation is used to validate in electrostatic potential and three dimensional laplace equation helps solve the short interest 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 dau squared psi upon dau x squared plus dau squared psi upon dau 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 dau squared psi upon dau x squared plus dau squared psi upon dau y squared plus dau squared psi upon dau 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 my psi, we obtain 1 upon x into d squared x upon d x squared plus 1 upon y into d squared y upon d y squared plus 1 upon z into d squared z upon d z 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 like, subscribe and comment with your feedback to help us make better videos. Thanks for watching. Also, thanks a lot for those constructive comments. You helped the channel grow. So here are the top mechanical e-highs of our last videos. In the next episode of Mechanical EI, find out what solid geometry is.