quality:correction, quality:corruption
Example 5

The Jacobian determinant of the function F : R → R with components y 1 = 5 x 2 y 2 = 4 x 1 2 − 2 sin (x 2 x 3) y 3 = x 2 x 3 {\displaystyle {\begin{aligned}y_{1}&=5x_{2}\\y_{2}&=4x_{1}^{2}-2\sin(x_{2}x_{3})\\y_{3}&=x_{2}x_{3}\end{aligned}}} is | 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 | = − 8 x 1 | 5 0 x 3 x 2 | = − 40 x 1 x 2. {\displaystyle {\begin{vmatrix}0&5&0\\8x_{1}&-2x_{3}\cos(x_{2}x_{3})&-2x_{2}\cos(x_{2}x_{3})\\0&x_{3}&x_{2}\end{vmatrix}}=-8x_{1}{\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}=-40x_{1}x_{2}.}

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