ABSTRACT
This chapter focuses on second order derivatives of functions of several variables. It discusses the Hessian matrix and its properties. As an application, the Laplace operator is introduced which is a key differential operator in mathematical modelling. The Chain Rule can be extended to second order derivatives as well. One important quantity that involves the second order derivatives is the Laplace operator (or simply Laplacian). The term Laplace operator was coined after the French mathematician Pierre-Simon de Laplace and has since been used in the mathematical modelling of various real life phenomena such as: heat and fluid flow through diffusion equation; electric and gravitational potentials through Poisson's equation; and the wave function in quantum mechanics through Schrödinger's equation. The field of mathematics that studies the harmonic functions is called Potential Theory.
