# Definitions for **jacobian**

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

### Wiktionary

Jacobian(Noun)

A Jacobian matrix or its associated operator.

Jacobian(Noun)

The determinant of such a matrix.

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

### Webster Dictionary

Jacobian(adj)

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.]

### Freebase

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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"jacobian." *Definitions.net.* STANDS4 LLC, 2015. Web. 31 Jul 2015. <http://www.definitions.net/definition/jacobian>.