1. We can also express this compactly using the Jacobian determinant : 2. The Jacobian determinant is occasionally referred to as " the Jacobian ". 3. Furthermore, if the Jacobian determinant at is negative, reverses orientation. 4. Note that the product of all the scale factors is the Jacobian determinant . 5. Such a function admits a Jacobian determinant . 6. The Jacobian determinant at a given point gives important information about the behavior of near that point. 7. We can then form its determinant, known as the "'Jacobian determinant " '. 8. The Jacobian determinant also appears when changing the variables in multiple integrals ( see substitution rule for multiple variables ). 9. For instance, the continuously differentiable function is invertible near a point if the Jacobian determinant at is non-zero. 10. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral.
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