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For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation. The absolute value of the ...
The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map , in which case it is a covering map of a simply connected manifold , hence invertible.
Lemma 1. ′ =, where ′ is the differential of . This equation means that the differential of det {\displaystyle \det } , evaluated at the identity matrix, is equal to the trace. The differential det ′ ( I ) {\displaystyle \det '(I)} is a linear operator that maps an n × n matrix to a real number.
Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the n × n matrices that has the four following properties: The determinant of the identity matrix is 1. The exchange of two rows multiplies the determinant by −1.
This means that the transition functions in the atlas of M are C 1-functions. Such a function admits a Jacobian determinant. When the Jacobian determinant is positive, the transition function is said to be orientation preserving. An oriented atlas on M is an atlas for which all transition functions are orientation preserving.
The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. [1]
However, the Levi-Civita symbol is a pseudotensor because under an orthogonal transformation of Jacobian determinant −1, for example, a reflection in an odd number of dimensions, it should acquire a minus sign if it were a tensor. As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor.
Top: The action of M, indicated by its effect on the unit disc D and the two canonical unit vectors e 1 and e 2. Left: The action of V ⁎, a rotation, on D, e 1, and e 2. Bottom: The action of Σ, a scaling by the singular values σ 1 horizontally and σ 2 vertically. Right: The action of U, another rotation.