# 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.

# Translation

#### Find a translation for the **jacobian** definition in other languages:

Select another language:

#### Discuss these jacobian definitions with the community:

# Word of the Day

#### Would you like us to send you a **FREE** new word definition delivered to your inbox daily?

# Citation

## Use the citation below to add this definition to your bibliography:

"jacobian." *Definitions.net.* STANDS4 LLC, 2015. Web. 4 Mar. 2015. <http://www.definitions.net/definition/jacobian>.