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\section{\label{Hessian}Matrix-Valued Operators}

Matrix-valued operators include the Hessian and the Jacobian.

\subsection{\ Hessian}

The 
\index{Hessian@Hessian} 
\index{Vector calculus@Vector calculus!Hessian@Hessian}\textsl{Hessian} is
the $n\times n\,$matrix \label{Vector calculus, Hessian} 
\[
\left[ 
\begin{array}{cccc}
\dfrac{\partial ^{2}f}{\partial x_{1}^{2}} & \dfrac{\partial ^{2}f}{\partial
x_{1}\partial x_{2}} & \cdots & \dfrac{\partial ^{2}f}{\partial
x_{1}\partial x_{n}} \\ 
\dfrac{\partial ^{2}f}{\partial x_{2}\partial x_{1}} & \dfrac{\partial ^{2}f%
}{\partial x_{2}^{2}} & \cdots & \dfrac{\partial ^{2}f}{\partial
x_{2}\partial x_{n}} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
\dfrac{\partial ^{2}f}{\partial x_{n}\partial x_{1}} & \dfrac{\partial ^{2}f%
}{\partial x_{n}\partial x_{2}} & \cdots & \dfrac{\partial ^{2}f}{\partial
x_{n}^{2}}
\end{array}
\right] \vspace{6pt} 
\]
of second partial derivatives of a scalar function $f\left(
x_{1},x_{2},\ldots ,x_{n}\right) \,$of $n$ variables. The order of the
variables affects the ordering of the rows and columns of the Hessian, but
it is natural to require that the list of variables be ordered
lexicographically, and this choice is the one made by \textsl{Scientific
Notebook}.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Vector Calculus +\ Hessian}
\end{quote}

\begin{quotation}
$xyz$, Hessian is $\left[ 
\begin{array}{ccc}
0 & z & y \\ 
z & 0 & x \\ 
y & x & 0
\end{array}
\right] \vspace{6pt}$

$x^{2}+y^{3}$, Hessian is $\left[ 
\begin{array}{cc}
2 & 0 \\ 
0 & 6y
\end{array}
\right] \vspace{6pt}$

$a^{3}+b^{3}$, Hessian is $\left[ 
\begin{array}{cc}
6a & 0 \\ 
0 & 6b
\end{array}
\right] \vspace{6pt}$

$b^{3}+a^{3}$, Hessian is $\left[ 
\begin{array}{cc}
6a & 0 \\ 
0 & 6b
\end{array}
\right] \vspace{6pt}$

$wxyz$,\thinspace Hessian\thinspace is\thinspace $\left[ 
\begin{array}{cccc}
0 & yz & xz & xy \\ 
yz & 0 & wz & wy \\ 
xz & wz & 0 & wx \\ 
xy & wy & wx & 0
\end{array}
\right] $\ \vspace{6pt}

$xzwy$,\thinspace Hessian\thinspace is\thinspace $\left[ 
\begin{array}{cccc}
0 & zy & xz & xy \\ 
zy & 0 & zw & wy \\ 
xz & zw & 0 & xw \\ 
xy & wy & xw & 0
\end{array}
\right] \medskip $
\end{quotation}

A possible source of confusion is the inclusion of something in the variable
list that was intended to be a constant parameter, such as $c$ in \textsl{%
cxyz}. In this case, $c$ will be interpreted as a variable, and you will
obtain a $4\times 4$ matrix. Since $c$ occurs first in the alphabet, partial
derivatives involving $c$ occur in the first row and column. Delete the
first row and first column of the resulting matrix to get the desired
result.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Vector Calculus +\ Hessian}
\end{quote}

\begin{quotation}
$cxyz$, Hessian is $\left[ 
\begin{array}{cccc}
0 & yz & xz & xy \\ 
yz & 0 & cz & cy \\ 
xz & cz & 0 & cx \\ 
xy & cy & cx & 0
\end{array}
\right] $\qquad or $\left[ 
\begin{array}{ccc}
0 & cz & cy \\ 
cz & 0 & cx \\ 
cy & cx & 0
\end{array}
\right] $ for $c$ constant\medskip
\end{quotation}

After defining $f(x,y,z)=3xy^2z$, you can find the Hessian for the
expression $f(x,y,z)$.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Vector Calculus + Hessian}
\end{quote}

\begin{quotation}
$f(x,y,z)$,\thinspace Hessian \thinspace is\thinspace $\left[ 
\begin{array}{ccc}
0 & 6yz & 3y^{2} \\ 
6yz & 6xz & 6xy \\ 
3y^{2} & 6xy & 0
\end{array}
\right] $\thinspace \vspace{6pt}

$f(z,y,x)$,\thinspace Hessian \thinspace is\thinspace $\left[ 
\begin{array}{ccc}
0 & 6zy & 3y^{2} \\ 
6zy & 6xz & 6xy \\ 
3y^{2} & 6xy & 0
\end{array}
\right] \vspace{6pt}$

$\left( f(y,z,x)\right) ^{2}$, Hessian is $\left[ 
\begin{array}{ccc}
18y^{2}z^{4} & 36yz^{4}x & 72y^{2}z^{3}x \\ 
36yz^{4}x & 18z^{4}x^{2} & 72yz^{3}x^{2} \\ 
72y^{2}z^{3}x & 72yz^{3}x^{2} & 108y^{2}z^{2}x^{2}
\end{array}
\right] \bigskip $
\end{quotation}

\subsection{\ Jacobian\label{Jacobian}}

The 
\index{Jacobian@Jacobian} 
\index{Vector calculus@Vector calculus!Jacobian@Jacobian}\textsl{Jacobian} 
\label{Vector calculus, Jacobian}is the $n\times n\,$matrix 
\[
\left[ 
\begin{array}{cccc}
\dfrac{\partial f_{1}}{\partial x_{1}} & \dfrac{\partial f_{1}}{\partial
x_{2}} & \cdots & \dfrac{\partial f_{1}}{\partial x_{n}} \\ 
\dfrac{\partial f_{2}}{\partial x_{1}} & \dfrac{\partial f_{2}}{\partial
x_{2}} & \cdots & \dfrac{\partial f_{2}}{\partial x_{n}} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
\dfrac{\partial f_{n}}{\partial x_{1}} & \dfrac{\partial f_{n}}{\partial
x_{2}} & \cdots & \dfrac{\partial f_{n}}{\partial x_{n}}
\end{array}
\right] 
\]
of partial derivatives of the entries in a vector field 
\[
\left( f_{1}\left( x_{1},x_{2},\ldots ,x_{n}\right) ,f_{2}\left(
x_{1},x_{2},\ldots ,x_{n}\right) ,\ldots ,f_{n}\left( x_{1},x_{2},\ldots
,x_{n}\right) \right) 
\]
Jacobians resemble Hessians in that the order of the variables in the
variable list determines the order of the columns of the matrix, and
lexicographic order is usually correct. The number of variables should be
the same as the dimension of the vector; if they are not the same, either a
parameter has been included in the variable list, or the vector field is
independent of one of the variables. In this case, a dialog box asks for the
list of variables. In each of the following examples, the variable list is $%
x,y,z$. To verify these examples, choose \textsf{Jacobian }while the
insertion point is in the given vector field.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Vector Calculus + Jacobian}
\end{quote}

\begin{quotation}
$(yz,xz,xy)$, Jacobian is $\left[ 
\begin{array}{ccc}
0 & z & y \\ 
z & 0 & x \\ 
y & x & 0
\end{array}
\right] \vspace{6pt}$

$(x^{2}z,x+z,xz^{2})$, Jacobian is $\left[ 
\begin{array}{ccc}
2xz & 0 & x^{2} \\ 
1 & 0 & 1 \\ 
z^{2} & 0 & 2xz
\end{array}
\right] \vspace{6pt}$\qquad ($y$ is missing)

$(x^{2}z,y+c,yz^{2})$, Jacobian is $\left[ 
\begin{array}{ccc}
2xz & 0 & x^{2} \\ 
0 & 1 & 0 \\ 
0 & z^{2} & 2yz
\end{array}
\right] $\qquad ($c$ is extra)\bigskip
\end{quotation}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Vectors}{}{}{DM9-1.tex#Vectors}

\item  \hyperref{Gradient, Divergence, and Curl}{}{}{DM9-2.tex#Gradient}

\item  \hyperref{Plots of Vector Fields and Gradients}{}{}{DM9-3.tex#Vector
field}

\item  \hyperref{Scalar and Vector Potentials}{}{}{DM9-4.tex#Scalar potential%
}

\item  \hyperref{Plots of Complex Functions}{}{}{DM9-6.tex#Conformal plot}

\item  \hyperref{Exercises and Solutions}{}{}{DM9-7.tex#Exercises}
\end{itemize}

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DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Vector calculus%
}{}{}{DMIndex.tex#Vector calculus}}

\end{document}
