Definitions for jacobian matrix and determinant
ja·co·bian ma·trix and de·ter·mi·nant

WikipediaRate this definition:0.0 / 0 votes

1. Jacobian matrix and determinant

   In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on Rn. This function takes a point x ∈ Rn as input and produces the vector f(x) ∈ Rm as output. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by J, whose (i,j)th entry is J i j = ∂ f i ∂ x j {\textstyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}} , or explicitly J = [ ∂ f ∂ x 1 ⋯ ∂ f ∂ x n ] = [ ∇ T f 1 ⋮ ∇ T f m ] = [ ∂ f 1 ∂ x 1 ⋯ ∂ f 1 ∂ x n ⋮ ⋱ ⋮ ∂ f m ∂ x 1 ⋯ ∂ f m ∂ x n ] {\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathrm {T} }f_{1}\\\vdots \\\nabla ^{\mathrm {T} }f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}} where ∇ T f i {\displaystyle \nabla ^{\mathrm {T} }f_{i}} is the transpose (row vector) of the gradient of the i {\displaystyle i} component. The Jacobian matrix, whose entries are functions of x, is denoted in various ways; common notations include Df, Jf, ∇ f {\displaystyle \nabla \mathbf {f} } , and ∂ ( f 1 , . . , f m ) ∂ ( x 1 , . . , x n ) {\displaystyle {\frac {\partial (f_{1},..,f_{m})}{\partial (x_{1},..,x_{n})}}} . Some authors define the Jacobian as the transpose of the form given above. The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix, the matrix product J(x) ⋅ h is another displacement vector, that is the best linear approximation of the change of f in a neighborhood of x, if f(x) is differentiable at x. This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. This linear function is known as the derivative or the differential of f at x. When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f. In particular, the function f has a differentiable inverse function in a neighborhood of a point x if and only if the Jacobian determinant is nonzero at x (see Jacobian conjecture for a related problem of global invertibility). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables). When m = 1, that is when f : Rn → R is a scalar-valued function, the Jacobian matrix reduces to the row vector ∇ T f {\displaystyle \nabla ^{\mathrm {T} }f} ; this row vector of all first-order partial derivatives of f is the transpose of the gradient of f, i.e. J f = ∇ T f {\displaystyle \mathbf {J} _{f}=\nabla ^{T}f} . Specializing further, when m = n = 1, that is when f : R → R is a scalar-valued function of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function f. These concepts are named after the mathematician Carl Gustav Jacob Jacobi (1804–1851).

WikidataRate this definition:0.0 / 0 votes

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, which is a slightly stronger condition than merely requiring that all partial derivatives exist there, then the derivative of F at p is the linear transformation represented by the matrix . This linear transformation is the best linear approximation of the function F near the point p. In the case the Jacobian matrix will be 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 and can be thought of as a local expansion factor for volumes; it is used when performing variable substitutions in multi-variable integrals, since it occurs prominently in the substitution rule for multiple variables.

How to pronounce jacobian matrix and determinant?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say jacobian matrix and determinant in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of jacobian matrix and determinant in Chaldean Numerology is: 7

2. Pythagorean Numerology

   The numerical value of jacobian matrix and determinant in Pythagorean Numerology is: 3

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Jacobian_Matrix_and_Determinant
2. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=jacobian matrix and determinant