Definitions for jacobian

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Wiktionary

  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

Freebase

  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 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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"jacobian." Definitions.net. STANDS4 LLC, 2014. Web. 20 Apr. 2014. <http://www.definitions.net/definition/jacobian>.

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