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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 Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p ; this is why it occurs in the general substitution rule .
According to van den Essen, [2] the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients. The obvious analogue of the Jacobian conjecture fails if k has characteristic p > 0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2.
The Jacobian of a function f(z) ... This is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix: ...
If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true ...
If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a. If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate). In those cases not listed above, the test is inconclusive. [2]
The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as = if is any vector whose sole non-zero entry is its first.
This means that if J is the Jacobian, then = and () =. Computing the Jacobian in the case z i = x i /‖ x ‖ 2, where ‖ x ‖ 2 = x 1 2 + ... + x n 2 gives JJ T = kI, with k = 1/‖ x ‖ 4n, and additionally det(J) is negative; hence the inversive map is anticonformal.
If the Jacobian of the dynamical system at an equilibrium happens to be a stability matrix (i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.