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Encyclopedia > Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. The French Revolution (1789–1815) was a period of political and social upheaval in the political history of France and Europe as a whole, during which the French governmental structure, previously an absolute monarchy with feudal privileges for the aristocracy and Catholic clergy, underwent radical change to forms based on... Jacobin may refer to: Members of the Jacobin Club, a political group during the French Revolution Jacobin (politics) and Jacobinism, pejorative epithets for left-wing revolutionary politics The term is unrelated to Jacobitism and the Jacobean era, both of which are related to the Stuart Dynasty in Great Britain. ... James II (14 October 1633 – 16 September 1701)[1] became King of England, King of Scots,[2] and King of Ireland on 6 February 1685. ... Charles Edward Stuart, Bonnie Prince Charlie, wearing the Jacobite blue bonnet Jacobitism was (and, to a very limited extent, remains) the political movement dedicated to the restoration of the Stuart kings to the thrones of England and Scotland. ... The term Jacobean refers to a period in English history that coincides with the reign of James I (1603 – 1625). ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...


In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded. Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the Jacobian variety of a non-singular algebraic curve C of genus g ≥ 1 is a particular abelian variety J, of dimension g. ... In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...


These concepts are all named after the mathematician Carl Gustav Jacobi. The term "Jacobian" is normally pronounced [jaˈkobiən], but can also be pronounced [ʤəˈkobiən]. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ...

Contents

Jacobian matrix

The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact that it represents the best linear approximation to a differentiable function near a given point. In this sense, the Jacobian is akin to a derivative of a multivariate function. For n > 1, the derivative of a numerical function must be matrix-valued, or a partial derivative. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...


Suppose F : RnRm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y1(x1,...,xn), ..., ym(x1,...,xn). The partial derivatives of all these functions (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix of F, as follows: Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

begin{bmatrix} frac{partial y_1}{partial x_1} & cdots & frac{partial y_1}{partial x_n}  vdots & ddots & vdots  frac{partial y_m}{partial x_1} & cdots & frac{partial y_m}{partial x_n} end{bmatrix}.

This matrix is denoted by

J_F(x_1,ldots,x_n) or by frac{partial(y_1,ldots,y_m)}{partial(x_1,ldots,x_n)}.

The ith row of this matrix is given by the transpose of the gradient of the function yi for i = 1,...,m. In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... For other uses, see Gradient (disambiguation). ...


If p is a point in Rn and F is differentiable at p, then its derivative is given by JF(p) (and this is the easiest way to compute the derivative). In this case, the linear map described by JF(p) is the best linear approximation of F near the point p, in the sense that For a non-technical overview of the subject, see Calculus. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ...

F(mathbf{x}) = F(mathbf{p}) + J_F(mathbf{p})(mathbf{x}-mathbf{p}) + o(|mathbf{x}-mathbf{p}|)

for x close to p and where o(...) is the little o-notation. For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...


Note that the Jacobian of the gradient is the Hessian matrix. In mathematics, the Hessian matrix is the square matrix of second order partial derivatives of a function. ...


If the components of F are arranged into a column vector In linear algebra, a column vector is an m × 1 matrix, i. ...

mathbf{y} = (y_1, dots, y_m)

the Jacobian may be represented as an outer product between the del operator and mathbf{y}: Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In vector calculus, del is a vector differential operator represented by the symbol ∇. This symbol is sometimes called the nabla operator, after the Greek word for a kind of harp with a similar shape (with related words in Aramaic and Hebrew). ...

frac{partial(y_1,ldots,y_m)}{partial(x_1,ldots,x_n)} = nabla otimes mathbf{y}

where the outer product symbol is often left out, it being understood that the gradient of a column vector is contextually a matrix. For other uses, see Gradient (disambiguation). ...


Examples

The transformation from spherical coordinates to Cartesian coordinates is given by the function F : R × [0,π] × [0,2π] → R3 with components: A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle... Fig. ...

 x_1 = r sinphi costheta ,
 x_2 = r sinphi sintheta ,
 x_3 = r cosphi ,

The Jacobian matrix for this coordinate change is

J_F(r,phi,theta) =begin{bmatrix} frac{partial x_1}{partial r} & frac{partial x_1}{partial phi} & frac{partial x_1}{partial theta} [3pt] frac{partial x_2}{partial r} & frac{partial x_2}{partial phi} & frac{partial x_2}{partial theta} [3pt] frac{partial x_3}{partial r} & frac{partial x_3}{partial phi} & frac{partial x_3}{partial theta}  end{bmatrix}=begin{bmatrix} sinphi costheta & r cosphi costheta & -r sinphi sintheta  sinphi sintheta & r cosphi sintheta & r sinphi costheta  cosphi & -r sinphi & 0 end{bmatrix}.

The Jacobian matrix of the function F : R3R4 with components

 y_1 = x_1 ,
 y_2 = 5x_3 ,
 y_3 = 4x_2^2 - 2x_3 ,
 y_4 = x_3 sin(x_1) ,

is

J_F(x_1,x_2,x_3) =begin{bmatrix} frac{partial y_1}{partial x_1} & frac{partial y_1}{partial x_2} & frac{partial y_1}{partial x_3} [3pt] frac{partial y_2}{partial x_1} & frac{partial y_2}{partial x_2} & frac{partial y_2}{partial x_3} [3pt] frac{partial y_3}{partial x_1} & frac{partial y_3}{partial x_2} & frac{partial y_3}{partial x_3} [3pt] frac{partial y_4}{partial x_1} & frac{partial y_4}{partial x_2} & frac{partial y_4}{partial x_3}  end{bmatrix}=begin{bmatrix} 1 & 0 & 0  0 & 0 & 5  0 & 8x_2 & -2  x_3cos(x_1) & 0 & sin(x_1) end{bmatrix}.

This example shows that the Jacobian need not be a square matrix.


In dynamical systems

Consider a dynamical system of the form x' = F(x), with F : RnRn. If F(x0) = 0, then x0 is a stationary point. The behavior of the system near a stationary point can often be determined by the eigenvalues of JF(x0), the Jacobian of F at the stationary point.[1] The Lorenz attractor is an example of a non-linear dynamical system. ...


Jacobian determinant

If m = n, then F is a function from n-space to n-space and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is also called the "Jacobian" in some sources. For the square matrix section, see square matrix. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...


The Jacobian determinant at a given point gives important information about the behavior of F near that point. For instance, the continuously differentiable function F is invertible near p if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then F preserves orientation near p; if it is negative, F reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function F expands or shrinks volumes near p; this is why it occurs in the general substitution rule. In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... For other uses, see Volume (disambiguation). ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...


Example

The Jacobian determinant of the function F : R3R3 with components

 y_1 = 5x_2 ,
 y_2 = 4x_1^2 - 2 sin (x_2x_3) ,
 y_3 = x_2 x_3 ,

is

begin{vmatrix} 0 & 5 & 0  8x_1 & -2x_3cos(x_2 x_3) & -2x_2cos(x_2 x_3)  0 & x_3 & x_2 end{vmatrix}=-8x_1cdotbegin{vmatrix} 5 & 0 x_3&x_2end{vmatrix}=-40x_1 x_2.

From this we see that F reverses orientation near those points where x1 and x2 have the same sign; the function is locally invertible everywhere except near points where x1 = 0 or x2 = 0. If you start with a tiny object around the point (1,1,1) and apply F to that object, you will get an object set with about 40 times the volume of the original one. In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ...


Uses

The Jacobian determinant is used when making a change of variables when integrating a function over its domain. To accommodate for the change of coordinates the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of coordinates is done in a manner which maintains an injectivity between the coordinates that determine the domain. The Jacobian determinant, as a result, is usually well defined. In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... This article is about the concept of integrals in calculus. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...


See also

Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. ... In mathematics, the Hessian matrix is the square matrix of second order partial derivatives of a function. ...

References

  1. ^ D.K. Arrowsmith and C.M. Place, Dynamical Systems, Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.

External links

  • Ian Craw's Undergraduate Teaching Page An easy to understand explanation of Jacobians
  • Mathworld A more technical explanation of Jacobians

  Results from FactBites:
 
PlanetMath: Jacobian matrix (278 words)
, it is easy to show that the effect of a change of coordinates on volume forms is a local scaling of the volume form by the determinant of the Jacobian matrix of the derivative of the backwards change of coordinates, which is called the inverse Jacobian.
The determinant of the inverse Jacobian is thus commonly seen in integration over a change of coordinates.
This is version 14 of Jacobian matrix, born on 2001-11-14, modified 2007-01-14.
Jacobian - Wikipedia, the free encyclopedia (640 words)
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.
The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
In this sense, the Jacobian is akin to a derivative of a multivariate function.
  More results at FactBites »

 
 

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