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The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: Jacobian matrix and determinant (and in particular, the robot Jacobian)
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.
That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points). The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If k, m, and n are 1, so that f : R → R and g : R → R, then the Jacobian matrices of f and g are 1 × 1.
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 ...
The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. [ 9 ] [ 10 ] A further generalization for a function between Banach spaces is the Fréchet derivative .
This coefficient matrix of the linear system is the Jacobian matrix (and its inverse) of the transformation. These are the equations that can be used to transform a Cartesian basis into a curvilinear basis, and vice versa. In three dimensions, the expanded forms of these matrices are
In particular, the Jacobian determinant of a bi-Lipschitz mapping det Dφ is well-defined almost everywhere. The following result then holds: The following result then holds: Theorem — Let U be an open subset of R n and φ : U → R n be a bi-Lipschitz mapping.