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The Jacobian determinant of a square-dimensional map is the determinant of its Jacobian matrix
Definition
Let , let be open, and let be . Its Jacobian matrix at is the matrix of the total derivative in the standard bases. Its Jacobian determinant is The change-of-variables scale factor is . Thus an orientation-reversing derivative and an orientation-preserving derivative with the same volume scale have the same change-of-variables factor.
Depends on
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- A total derivative computes every directional derivative, and its matrix is the Jacobian
- For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
Used by
- Polar coordinates on a full closed angular period are not injective and are singular at radius zero Counterexample
- Local orientation of a regular C¹ Euclidean map Definition
- Surface reparametrizations and their orientation sign Definition
- Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
- Polar change of variables on a compact annular sector gives the Jacobian factor r and its area Example
- Spherical coordinates have absolute Jacobian determinant r² sinφ away from the axis and angular seam Example
- The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
- FALSE: spherical coordinates are globally injective False statement
- A cyclic permutation of the coordinates of ℝ³ preserves Jordan measurability and integrals Lemma
- Change of variables for a C¹ map injective and regular only on the interior of a compact Jordan set Lemma
- Each coordinate of the oriented area vector is the Jacobian determinant of the matching cyclic projection Lemma
- On a small cube, a C¹ diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant Lemma
- The plane Gaussian integral equals π by polar coordinates Lemma
- At interior base points, the graph faces of an adapted presentation induce the outward unit normal Proposition
- Change of variables for an injective C¹ map on a compact Jordan set Theorem
Dependency tree · two levels
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Sources
- A. Leibman, Multidimensional Real Analysis, §5.5 (standard reference, not scraped)