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Jacobian matrix and determinantFrom Wikipedia, the free encyclopedia

In vector calculus, the Jacobian matrix (/dʒɨ koʊbiәn/, /jɨˈkoʊbiәn/) is the matrix of all first-order partial

derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its

determinant are r eferred to as the Jacobian in literature.[1]

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 , Jf , and∂(f 1,...,f m)

∂(x1,...,xn). (Note that some

literature defines the Jacobian as the transpose of the matrix given above.)

The Jacobian matrix is important because if the function f is differentiable at a point x (this is a slightly

stronger condition than merely requiring that all partial derivatives exist there), then the Jacobian matrix

defines a linear mapℝn →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 Jaco bian 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, th

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 (see Jacobian conjecture). The Jacobian determinant occurs als

when changing the variables in multi-variable integrals (see substitution rule for multiple variables).

If m = 1, f is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of f —

i.e. the gradient of f .

These concepts are named after the mathematician Carl Gustav Jacob Jacobi (1804–1851).

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Contents

1 Jacobian matrix

2 Jacobian determinant

3 Inverse

4 Critical points

5 Examples

5.1 Example 1

5.2 Example 2: polar-Cartesian transformation

5.3 Example 3: spherical-Cartesian transformation

5.4 Example 4

5.5 Example 5

6 Other uses

6.1 Dynamical systems

6.2 Newton's method

7 See also

8 References

9 Further reading

10 External links

Jacobian matrix

The Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself

generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for

scalar-valued multivariable function is the gradient and that of a scalar-valued function of single variable i

simply its derivative. The Jacobian can also be thought of as describing the amount of "stretching",

"rotating" or "transforming" that a transformation imposes locally. For example, if ( x′, y′) = f ( x, y) is use

to transform an image, the Jacobian Jf ( x, y), describes how the image in the neighborhood of ( x, y) is

transformed.

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A nonlinear map f : R 2 → R 2 sends a small square to a distorted

parallelepiped close to the image of the square under the best linear

approximation of f near the point.

If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function

doesn't need to be differentiable for the Jacobian to be defined, since only the partial derivatives are

required to exist.

If p is a point inℝn and f is differentiable at p, then its derivative is given by Jf (p). In this case, the linea

map described by Jf (p) is the best linear approximation of f near the point p, in the sense that

for x close to p and where o is the little o-notation (for x → p) and ‖x − p‖ is the distance between x and

p.

Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order:

In a sense, both the gradient and Jacobian are "first derivatives"—the former the first derivative of a scalar

unction of several variables, the latter the first derivative of a vector function of several variables.

The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian

matrix, which in a sense is the "second derivative" of the function in question.

Jacobian determinant

If m=n, then f is a function fromℝn to

itself and the Jacobian matrix is a square

matrix. We can then form its

determinant, known as the Jacobiandeterminant. The Jacobian determinant

is occasionally referred to as "the

Jacobian".

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 a point

p ∈ ℝn if the Jacobian determinant atp 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 u

the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the general

substitution rule.

The Jacobian determinant is used when making a change of variables when evaluating a multiple integral o

a function over a region within its domain. To accommodate for the change of coordinates the magnitude o

the Jacobian determinant arises as a multiplicative factor within the integral. This is because the n-

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dimensional dV element is in general a parallelepiped in the new coordinate system, and the n-volume of

parallelepiped is the determinant of its edge vectors.

The Jacobian can also be used to solve systems of differential equations at an equilibrium point or

approximate solutions near an equilibrium point.

Inverse

According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible

function is the Jacobian matrix of the inverse function. That is, if the Jacobian of the function

f :ℝn →ℝn is continuous and nonsingular at the point p inℝn, then f is invertible when restricted to

some neighborhood of p and

Conversely, if the Jacobian determinant is not zero at a point, then the function is locally invertible near th

point, that is there is neighbourhood of this point, in which the function is invertible.

The (unproved) Jacobian conjecture is related to global invertibility in the case of a polynomial functions,

that is a function defined by n polynomials in n variables. It asserts that, if the Jacobian determinant is a

non-zero constant (or, equivalently, that it does not have any complex zero), then the function is invertible

and its inverse is a polynomial function.

Critical points

If f :ℝn →ℝm is a differentiable function, a critical point of f is a point where the rank of the Jacobian

matrix is not maximal. This means that the rank at the critical point is lower than the rank at some

neighbour point. In other words, let k be the maximal dimension of the open balls contained in the image o

f ; then a point is critical if all minors of rank k of f are zero.

In the case where 1 = m = n = k , a point is critical if the Jacobian determinant is zero.

Examples

Example 1

Consider the function f :ℝ2 →ℝ2 given by

Then we have

and

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and the Jacobian matrix of F is

and the Jacobian determinant is

Example 2: polar-Cartesian transformation

The transformation from polar coordinates (r , φ) to Cartesian coordinates ( x, y), is given by the function

F:ℝ

+

× [0, 2 π ) →ℝ

2

with components:

The Jacobian determinant is equal to r . This can be used to transform integrals between the two coordinatesystems:

Example 3: spherical-Cartesian transformation

The transformation from spherical coordinates (r , θ , φ) to Cartesian coordinates ( x, y, z ), is given by the

functionF:ℝ

+

× [0, π ] × [0, 2 π ) →ℝ

3 with components:

The Jacobian matrix for this coordinate change is

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The determinant is r 2 sin θ . As an example, since dV = dx1 dx2 dx3 this determinant implies that the

differential volume element dV = r 2 sin θ dr dθ dφ. Nevertheless this determinant varies with

coordinates.

Example 4

The Jacobian matrix of the function F :ℝ3 →ℝ4 with components

is

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

Example 5

The Jacobian determinant of the function F :ℝ3 →ℝ3 with components

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is

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. Intuitively, if one

starts with a tiny object around the point (1, 2, 3) and apply F to that object, one will get a resulting obje

with approximately 40 × 1 × 2 = 80 times the volume of the original one.

Other uses

The Jacobian serves as a linearized design matrix in statistical regression and curve fitting; see non-linear

least squares.

Dynamical systems

Consider a dynamical system of the form x′ = F(x), where x′ is the (component-wise) time derivative of

x, and F :ℝn →ℝn is differentiable. If F(x0) = 0, then x0 is a stationary point (also called a critical

point; this is not to be confused with fixed points). The behavior of the system near a stationary point is

related to the eigenvalues of JF(x0), the Jacobian of F at the stationary point.[2] Specifically, if the

eigenvalues all have real parts that are negative, then the system is stable near the stationary point, if any

eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the

eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.

Newton's method

A system of coupled nonlinear equations can be solved iteratively by Newton's method. This method uses

the Jacobian matrix of the system of equations.

See also

Hessian matrixPushforward (differential)

References

1. Mathworld (http://mathworld.wolfram.com/Jacobian.html)

2. Arrowsmith, D. K.; Place, C. M. (1992). "Section 3.3". Dynamical Systems. London: Chapman & Hall. ISBN 0

412-39080-9.

Further reading

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Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 305–330. ISBN 540-60988-1.

External links

Hazewinkel, Michiel, ed. (2001), "Jacobian", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Mathworld (http://mathworld.wolfram.com/Jacobian.html) A more technical explanation of Jacobians

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Categories: Multivariable calculus Differential calculus Generalizations of the derivative

Determinants Matrices

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