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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 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 .
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 ]
The abelian variety J is called the Jacobian variety of C, for any non-singular curve C over the complex numbers. From the point of view of birational geometry , its function field is the fixed field of the symmetric group on g letters acting on the function field of C g {\displaystyle C^{g}} .
In much the same way, one can define a graph-theoretic analogue of Abel–Jacobi map as a piecewise-linear map from a finite graph into a flat torus (or a Cayley graph associated with a finite abelian group), which is closely related to asymptotic behaviors of random walks on crystal lattices, and can be used for design of crystal structures.
Plot of the Jacobi polynomial function (,) with = and = and = in the complex plane from to + with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) (,) are a class of classical orthogonal polynomials.
Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.