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
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 is a function. Such a function is given by m real-valued component functions, . The partial derivatives of all these functions with respect to the variables can be organized in an m-by-n matrix, the Jacobian matrix of, as follows: This matrix, whose entries are functions of, is also denoted by and . The Jacobian matrix is important because if the function F is differentiable at a point, then the Jacobian matrix defines a linear map, which is the best linear approximation of the function F near the point p. This linear map is thus the generalization of the usual notion of derivative, and is called the derivative or the differential of F at p. In the case the Jacobian matrix is a square matrix, and its determinant, a function of, 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 p an inverse function that is differentiable if and only if the Jacobian determinant is nonzero at p. The Jacobian determinant occurs also when changing the variables in multi-variable integrals.
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