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In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
The polynomial x − x p has derivative 1 − p x p−1 which is 1 (because px is 0) but it has no inverse function. However, Kossivi Adjamagbo suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F). [3]
[1] Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind ...
The SR1 formula does not guarantee the update matrix to maintain positive-definiteness and can be used for indefinite problems. The Broyden's method does not require the update matrix to be symmetric and is used to find the root of a general system of equations (rather than the gradient) by updating the Jacobian (rather than the Hessian).
For functions of more than one variable, the theorem states that if is a continuously differentiable function from an open subset of into , and the derivative ′ is invertible at a point a (that is, the determinant of the Jacobian matrix of f at a is non-zero), then there exist neighborhoods of in and of = such that () and : is bijective. [1]
The Jacobian matrix, J, is a function of constants, ... 0.001 is equivalent to specifying that each parameter should be refined to 0.1% precision. This is reasonable ...
The same terminology applies. A regular solution is a solution at which the Jacobian is full rank (). A singular solution is a solution at which the Jacobian is less than full rank. A regular solution lies on a k-dimensional surface, which can be parameterized by a point in the tangent space (the null space of the Jacobian).
In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C , hence an abelian variety .