Editing Jacobian matrix and determinant (section)

=== Example 5 ===

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

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

is

<math display="block">\begin{vmatrix}
  0 & 5 & 0 \\
  8 x_1 & -2 x_3 \cos(x_2 x_3) & -2 x_2 \cos (x_2 x_3) \\
  0 & x_3 & x_2
\end{vmatrix} = -8 x_1 \begin{vmatrix}
  5 & 0 \\
  x_3 & x_2
\end{vmatrix} = -40 x_1 x_2.</math>

From this we see that {{math|'''F'''}} reverses orientation near those points where {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} have the same sign; the function is [[locally]] invertible everywhere except near points where {{math|''x''<sub>1</sub> {{=}} 0}} or {{math|''x''<sub>2</sub> {{=}} 0}}. Intuitively, if one starts with a tiny object around the point {{math|(1, 2, 3)}} and apply {{math|'''F'''}} to that object, one will get a resulting object with approximately {{math|40 × 1 × 2 {{=}} 80}} times the volume of the original one, with orientation reversed.