The Hessian Matrix: Derivation, Interpretation, and Example, Real Analysis II

In this lecture, we explore higher-order differentiation by rigorously deriving and interpreting the Hessian matrix for scalar-valued functions of several variables. We begin by reviewing the Jacobian matrix and its interpretation as a linear transformation that computes first-order directional derivatives. Building on that foundation, we expand the gradient in coordinates and apply second-order differentiation to construct the Hessian matrix. We emphasize the symmetry of the Hessian matrix under the assumption that the function is class C² (as ensured by Clairaut’s theorem), and interpret the Hessian as a bilinear form that measures the behavior of iterated directional derivatives. Through further discussion of its geometric significance, we situate the Hessian within the broader framework of differentiation and hint at its extension to vector-valued functions and higher-order tensors. Proof of Clairaut's theorem (so that Hf is symmetric whenever f is C2) is here: https://youtu.be/nQ5BkJ9yNVw #mathematics #math #realanalysis #jacobian #differentiation #vectorcalculus