The (unproved) Jacobian conjecture is related to global invertibility in the case of a polynomial function, that is a function defined by n polynomials in n variables. In other words, if the Jacobian determinant is not zero at a point, then the function is locally invertible near this point, that is, there is a neighbourhood of this point in which the function is invertible. Then the Jacobian matrix of f is defined to be an m× n matrix, denoted by J, whose ( i, j)th entry is J i j = ∂ f i ∂ x j
This function takes a point x ∈ R n as input and produces the vector f( x) ∈ R m as output. Suppose f : R n → R m is a function such that each of its first-order partial derivatives exist on R n. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. In vector calculus, the Jacobian matrix ( / dʒ ə ˈ k oʊ b i ə n/, / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
