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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 .
In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n -dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.
The Jacobian also occurs in the inverse function theorem. When applied to the field of Cartography , the determinant can be used to measure the rate of expansion of a map near the poles. [ 41 ]
Let J p (v) be the n×n Jacobian matrix of the vector field v at the point p. If all eigenvalues of J have strictly negative real part then the solution is asymptotically stable. This condition can be tested using the Routh–Hurwitz criterion.
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 ...
This formula gives R as the sum of the sines of four non-negative angles with sum 2π, so it is always non-negative. [1] But then the Jacobian at 0 is strictly positive and F f is therefore locally a diffeomorphism. It remains to deduce F f is a homeomorphism. By continuity its image is compact so closed.
The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety , of dimension g , and hence, over the complex numbers, it is a complex torus .
In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let (, ...