In

vector calculus, the

**Jacobian matrix** (icon, rarely j) is the

matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

of all first-order

partial derivativeIn 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...

s of a

vectorA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

- or scalar-valued

functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

with respect to another vector. Suppose

*F* :

**R**^{n} →

**R**^{m} is a function from

Euclidean *n*-spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

to Euclidean

*m*-space. Such a function is given by

*m* real-valued component functions,

*y*_{1}(

*x*_{1},...,

*x*_{n}), ...,

*y*_{m}(

*x*_{1},...,

*x*_{n}). The partial derivatives of all these functions (if they exist) can be organized in an

*m*-by-

*n* matrix, the Jacobian matrix

*J* of

*F*, as follows:

This matrix is also denoted by

and

. If (

*x*_{1},...,

*x*_{n}) are the usual orthogonal Cartesian coordinates, the

*i* th row (

*i* = 1, ...,

*n*) of this matrix corresponds to the

gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of the

*i*^{th} component function

*y*_{i}:

. Note that some books define the Jacobian as the transpose of the matrix given above.

The

**Jacobian determinant** (often simply called the

**Jacobian**) is the

determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of the Jacobian matrix (if

).

These concepts are named after the

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Carl Gustav Jacob Jacobi.

## Jacobian matrix

The Jacobian of a function describes the

orientationIn mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

of a tangent plane to the function at a given point. In this way, the Jacobian generalizes the

gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of a scalar valued function of multiple variables which itself generalizes the derivative of a scalar-valued function of a scalar. Likewise, the Jacobian can also be thought of as describing the amount of "stretching" that a transformation imposes. For example, if

is used to transform an image, the Jacobian of

*f*,

describes how much the image in the neighborhood of

is stretched in the

*x* and

*y* directions.

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 derivativeIn 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...

s are required to exist.

The importance of the Jacobian 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 the derivative of a multivariate function.

If

**p** is a point in

**R**^{n} and

*F* is

differentiableIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

at

**p**, then its derivative is given by

*J*_{F}(

**p**). In this case, the linear map described by

*J*_{F}(

**p**) is the best

linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...

of

*F* near the point

**p**, in the sense that

for

**x** close to

**p** and where

*o* is the little o-notation (for

) and

is the

distanceIn mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

between

**x** and

**p**.

In a sense, both the

gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

and Jacobian are "

first derivativesIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

" the former the first derivative of a

*scalar function* of several variables, the latter the first derivative of a

*vector function* of several variables. In general, the

gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

can be regarded as a special version of the Jacobian: it is the Jacobian of a scalar function of several variables.

The Jacobian of the

gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

has a special name: the

Hessian matrixIn mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...

, which in a sense is the "

second derivativeIn calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...

" of the scalar function of several variables in question.

### Inverse

According to the

inverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...

, the

matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the

*inverse* function. That is, for some function

*F* :

**R**^{n} →

**R**^{n} and a point

*p* in

**R**^{n},

It follows that the (scalar) inverse of the Jacobian determinant of a transformation is the Jacobian determinant of the inverse transformation.

#### Dynamical systems

Consider a

dynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

of the form

*x*' =

*F*(

*x*), where

*x*' is the (component-wise) time derivative of

*x*, and

*F* :

**R**^{n} →

**R**^{n} is continuous and differentiable. If

*F*(

*x*_{0}) = 0, then

*x*_{0} is a stationary point (also called a fixed point). The behavior of the system near a stationary point is related to the eigenvalues of

*J*_{F}(

*x*_{0}), the Jacobian of

*F* at the stationary point. Specifically, if the eigenvalues all have a negative real part, then the system is stable in the operating point, if any eigenvalue has a positive real part, then the point is unstable.

#### 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.

## Jacobian determinant

If

*m* =

*n*, then

*F* is a function from

*m*-space to

*n*-space and the Jacobian matrix is a square matrix. We can then form its

determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

, known as the

**Jacobian determinant**. The Jacobian determinant is sometimes simply called "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** ∈

**R**^{n} if the Jacobian determinant at

**p** is non-zero. This is the

inverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...

. Furthermore, if the Jacobian determinant at

**p** is positive, then

*F* preserves orientation near

**p**; if it is negative,

*F* reverses orientation. The

absolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of the Jacobian determinant at

**p** gives us the factor by which the function

*F* expands or shrinks

volumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

s near

**p**; this is why it occurs in the general substitution rule.

### Uses

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

multiple integralThe multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...

of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of coordinates be 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.

## Examples

**Example 1.** The transformation from

spherical coordinatesIn mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...

(

*r*,

*θ*,

*φ*) to

Cartesian coordinatesA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

(

*x*_{1},

*x*_{2},

*x*_{3}) , is given by the function

*F* :

**R**^{+} × [0,π] × [0,2π) →

**R**^{3} with components:

The Jacobian matrix for this coordinate change is

The

determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

is

*r*^{2} sin

*θ*. As an example, since

*dV* =

*dx*_{1} *dx*_{2} *dx*_{3} this determinant implies that the differential volume element

*dV* =

*r*^{2} sin

*θ* *dr* *dθ* *dϕ*. Nevertheless this determinant varies with coordinates. To avoid any variation the new coordinates can be defined as

Now the determinant equals to 1 and volume element becomes

.

**Example 2.** The Jacobian matrix of the function

*F* :

**R**^{3} →

**R**^{4} with components

is

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

**Example 3.**
The Jacobian determinant is equal to

.

This shows how an integral in the

Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

is transformed into an integral in the

polar coordinate systemIn mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

:

.

**Example 4.**
The Jacobian determinant of the function

*F* :

**R**^{3} →

**R**^{3} with components

is

From this we see that

*F* reverses orientation near those points where

*x*_{1} and

*x*_{2} have the same sign; the function is locally invertible everywhere except near points where

*x*_{1} = 0 or

*x*_{2} = 0. Intuitively, 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 approximately 40 times the volume of the original one.

## External links

- Mathworld A more technical explanation of Jacobians