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