\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \)

Jacobian and Hessian

A Jacobian matrix collects all of the first partial derivatives of a function with multiple inputs and multiple outputs. The columns of the Jacobian keep the [input or output?] constant, and the rows keep the [ input or output?] constant. So, a function with a Jacobian matrix that is a colomun vector must be a Jacobian for a function with a single [input or output?].

Extending the idea of the Jacobian to second partial derivatives would give us a 3D tensor. A special case is when a function has only 1 output: In this case, the collection of all second partial derivatives can be collected in a 2D matrix. This is what is called the Hessian.