Definitions for jacobian
This page provides all possible meanings and translations of the word jacobian
A Jacobian matrix or its associated operator.
The determinant of such a matrix.
Origin: After Carl Gustav Jakob Jacobi, a German mathematician of the 19th century
of or pertaining to a style of architecture and decoration in the time of James the First, of England
Origin: [From L. Jacobus James. See 2d Jack.]
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 which takes as input the vector x ∈ ℝⁿ and produces as output the vector F(x) ∈ ℝᵐ. Then the Jacobian matrix J of F is an m×n matrix, usually defined and arranged as follows: or, component-wise: This matrix, whose entries are functions of x, is also denoted by DF, 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. If 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. In particular, the function F has locally in the neighborhood of a point x an inverse function that is differentiable if and only if the Jacobian determinant is nonzero at x.
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