Definitions for jacobian

This page provides all possible meanings and translations of the word jacobian


  1. Jacobian(Noun)

    A Jacobian matrix or its associated operator.

  2. Jacobian(Noun)

    The determinant of such a matrix.

  3. Origin: After Carl Gustav Jakob Jacobi, a German mathematician of the 19th century

Webster Dictionary

  1. Jacobian(adj)

    of or pertaining to a style of architecture and decoration in the time of James the First, of England


  1. Jacobian matrix and determinant

    In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Specifically, suppose F : ℝⁿ → ℝᵐ is a function. Such a function is given by m real-valued component functions, F₁, …, Fₘ. The partial derivatives of all these functions with respect to the variables x₁, …, xₙ can be organized in an m-by-n matrix, the Jacobian matrix J of F, as follows: This matrix, whose entries are functions of x₁, …, xₙ, is also denoted by J and . The Jacobian matrix is important because if the function F is differentiable at a point x =, then the Jacobian matrix defines a linear map ℝⁿ → ℝᵐ, which is the best linear approximation of the function F near the point x. This linear map is thus the generalization of the usual notion of derivative, and is called the derivative or the differential of F at x. In the case m = n the Jacobian matrix is a square matrix, and its determinant, a function of x₁, …, xₙ, is the Jacobian determinant of F. It carries important information about the local behavior of F.


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