Jacobian

The Jacobian is the n×n matrix

$\displaystyle \left[\vphantom{
\begin{array}{cccc}
\dfrac{\partial f_{1}}{\p...
...x_{2}} & \cdots & \dfrac{\partial f_{n}}{\partial x_{n}}
\end{array}
}\right.$$\displaystyle \begin{array}{cccc}
\dfrac{\partial f_{1}}{\partial x_{1}} & \df...
...\partial
x_{2}} & \cdots & \dfrac{\partial f_{n}}{\partial x_{n}}
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cccc}
\dfrac{\partial f_{1}}{\p...
...x_{2}} & \cdots & \dfrac{\partial f_{n}}{\partial x_{n}}
\end{array}
}\right]$

of partial derivatives of the entries in a vector field

$\displaystyle \left(\vphantom{ f_{1}\left( x_{1},x_{2},\ldots ,x_{n}\right) ,f_...
...s ,x_{n}\right) ,\ldots ,f_{n}\left( x_{1},x_{2},\ldots
,x_{n}\right) }\right.$f1$\displaystyle \left(\vphantom{ x_{1},x_{2},\ldots ,x_{n}}\right.$x1, x2,…, xn$\displaystyle \left.\vphantom{ x_{1},x_{2},\ldots ,x_{n}}\right)$, f2$\displaystyle \left(\vphantom{ x_{1},x_{2},\ldots ,x_{n}}\right.$x1, x2,…, xn$\displaystyle \left.\vphantom{ x_{1},x_{2},\ldots ,x_{n}}\right)$,…, fn$\displaystyle \left(\vphantom{ x_{1},x_{2},\ldots ,x_{n}}\right.$x1, x2,…, xn$\displaystyle \left.\vphantom{ x_{1},x_{2},\ldots ,x_{n}}\right)$$\displaystyle \left.\vphantom{ f_{1}\left( x_{1},x_{2},\ldots ,x_{n}\right) ,f_...
...s ,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 Jacobian while the insertion point is in the given vector field.


$\blacktriangleright$ Vector Calculus + Jacobian

(yz, xz, xy), Jacobian 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

(x2z, x + z, xz2), Jacobian is $\left[\vphantom{
\begin{array}{ccc}
2xz & 0 & x^{2} \\
1 & 0 & 1 \\
z^{2} & 0 & 2xz
\end{array}
}\right.$$\begin{array}{ccc}
2xz & 0 & x^{2} \\
1 & 0 & 1 \\
z^{2} & 0 & 2xz
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
2xz & 0 & x^{2} \\
1 & 0 & 1 \\
z^{2} & 0 & 2xz
\end{array}
}\right]$ 6pt        (y is missing)

(x2z, y + c, yz2), Jacobian is $\left[\vphantom{
\begin{array}{ccc}
2xz & 0 & x^{2} \\
0 & 1 & 0 \\
0 & z^{2} & 2yz
\end{array}
}\right.$$\begin{array}{ccc}
2xz & 0 & x^{2} \\
0 & 1 & 0 \\
0 & z^{2} & 2yz
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
2xz & 0 & x^{2} \\
0 & 1 & 0 \\
0 & z^{2} & 2yz
\end{array}
}\right]$        (c is extra)