Understanding the Laplacian and the Harmonic Functions
There are a plethora of important operators in mathematics that every engineer and physicist must know. We have already talked about the divergence and the curl in previous articles. Here, we are going to try to develop a deep understanding of the Laplacian, an operator which pops up all the time in physics problems. But what exactly is the Laplacian and how does it correlate with the Harmonic functions?
The Laplacian
In simple words, an operator in mathematics is an entity that takes as input a function and gives us another function as an output.
Imagine we have a scalar function of two variables, f = f(x,y).
The Laplacian of f = f(x,y) is defined as the divergence of its gradient which is equal to the sum of the function’s second spatial derivatives in cartesian coordinates.
The symbol we usually use to denote the Laplacian is either the del operator squared, ∇², or an upside triangle, Δ.
Note that this is the form of the Laplacian only in cartesian coordinates. When we use other coordinate systems such as polar, cylindrical, or spherical, the…
