Editing Jacobian matrix and determinant (section)

=== Example 4 ===

The Jacobian matrix of the function {{math|'''F''' : '''R'''<sup>3</sup> → '''R'''<sup>4</sup>}} with components

<math display="block">\begin{align}
y_1 &= x_1 \\
y_2 &= 5 x_3 \\
y_3 &= 4 x_2^2 - 2 x_3 \\
y_4 &= x_3 \sin x_1
\end{align}</math>

is

<math display="block">\mathbf J_{\mathbf F}(x_1, x_2, x_3) = \begin{bmatrix}
  \dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \dfrac{\partial y_1}{\partial x_3} \\[1em]
  \dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \dfrac{\partial y_2}{\partial x_3} \\[1em]
  \dfrac{\partial y_3}{\partial x_1} & \dfrac{\partial y_3}{\partial x_2} & \dfrac{\partial y_3}{\partial x_3} \\[1em]
  \dfrac{\partial y_4}{\partial x_1} & \dfrac{\partial y_4}{\partial x_2} & \dfrac{\partial y_4}{\partial x_3} \end{bmatrix}
= \begin{bmatrix}
  1 & 0 & 0 \\
  0 & 0 & 5 \\
  0 & 8 x_2 & -2 \\
  x_3\cos x_1 & 0 & \sin x_1 \end{bmatrix}.</math>

This example shows that the Jacobian matrix need not be a square matrix.