Hessian

The Hessian is the n×n matrix

$\displaystyle \left[\vphantom{
\begin{array}{cccc}
\dfrac{\partial ^{2}f}{\p...
...} & \cdots & \dfrac{\partial ^{2}f}{\partial
x_{n}^{2}}
\end{array}
}\right.$$\displaystyle \begin{array}{cccc}
\dfrac{\partial ^{2}f}{\partial x_{1}^{2}} &...
...tial x_{2}} & \cdots & \dfrac{\partial ^{2}f}{\partial
x_{n}^{2}}
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cccc}
\dfrac{\partial ^{2}f}{\p...
...} & \cdots & \dfrac{\partial ^{2}f}{\partial
x_{n}^{2}}
\end{array}
}\right]$ 6pt

of second partial derivatives of a scalar function f$\left(\vphantom{
x_{1},x_{2},\ldots ,x_{n}}\right.$x1, x2,…, xn$\left.\vphantom{
x_{1},x_{2},\ldots ,x_{n}}\right)$ 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 Scientific Notebook.


$\blacktriangleright$ Vector Calculus + Hessian

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

x2 + y3, Hessian is $\left[\vphantom{
\begin{array}{cc}
2 & 0 \\
0 & 6y
\end{array}
}\right.$$\begin{array}{cc}
2 & 0 \\
0 & 6y
\end{array}$$\left.\vphantom{
\begin{array}{cc}
2 & 0 \\
0 & 6y
\end{array}
}\right]$ 6pt

a3 + b3, Hessian is $\left[\vphantom{
\begin{array}{cc}
6a & 0 \\
0 & 6b
\end{array}
}\right.$$\begin{array}{cc}
6a & 0 \\
0 & 6b
\end{array}$$\left.\vphantom{
\begin{array}{cc}
6a & 0 \\
0 & 6b
\end{array}
}\right]$ 6pt

b3 + a3, Hessian is $\left[\vphantom{
\begin{array}{cc}
6a & 0 \\
0 & 6b
\end{array}
}\right.$$\begin{array}{cc}
6a & 0 \\
0 & 6b
\end{array}$$\left.\vphantom{
\begin{array}{cc}
6a & 0 \\
0 & 6b
\end{array}
}\right]$ 6pt

wxyz,Hessianis $\left[\vphantom{
\begin{array}{cccc}
0 & yz & xz & xy \\
yz & 0 & wz & wy \\
xz & wz & 0 & wx \\
xy & wy & wx & 0
\end{array}
}\right.$$\begin{array}{cccc}
0 & yz & xz & xy \\
yz & 0 & wz & wy \\
xz & wz & 0 & wx \\
xy & wy & wx & 0
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
0 & yz & xz & xy \\
yz & 0 & wz & wy \\
xz & wz & 0 & wx \\
xy & wy & wx & 0
\end{array}
}\right]$

xzwy,Hessianis $\left[\vphantom{
\begin{array}{cccc}
0 & zy & xz & xy \\
zy & 0 & zw & wy \\
xz & zw & 0 & xw \\
xy & wy & xw & 0
\end{array}
}\right.$$\begin{array}{cccc}
0 & zy & xz & xy \\
zy & 0 & zw & wy \\
xz & zw & 0 & xw \\
xy & wy & xw & 0
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
0 & zy & xz & xy \\
zy & 0 & zw & wy \\
xz & zw & 0 & xw \\
xy & wy & xw & 0
\end{array}
}\right]$


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 cxyz. In this case, c will be interpreted as a variable, and you will obtain a 4×4 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.


$\blacktriangleright$ Vector Calculus + Hessian

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


After defining f (x, y, z) = 3xy2z, you can find the Hessian for the expression f (x, y, z).


$\blacktriangleright$ Vector Calculus + Hessian

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

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

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