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In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdots & \dfrac \end where \nabla^ f_i is the transpose (row vector) of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the i component. The Jacobian matrix, whose entries are functions of , is denoted in various ways; common notations include , , \nabla \mathbf, and \frac. Some authors define the Jacobian as the transpose of the form given above. The Jacobian matrix represents the differential of at every point where is differentiable. In detail, if is a displacement vector represented by a
column matrix In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
, the matrix product is another displacement vector, that is the best linear approximation of the change of in a neighborhood of , if is differentiable at . This means that the function that maps to is the best linear approximation of for all points close to . This linear function is known as the ''derivative'' or the ''differential'' of at . When , the Jacobian matrix is square, so its
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
is a well-defined function of , known as the Jacobian determinant of . It carries important information about the local behavior of . In particular, the function has a differentiable inverse function in a neighborhood of a point if and only if the Jacobian determinant is nonzero at (see
Jacobian conjecture In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero c ...
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 , that is when is a scalar-valued function, the Jacobian matrix reduces to the row vector \nabla^ f; this row vector of all first-order partial derivatives of is the transpose of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of , i.e. \mathbf_ = \nabla^T f . Specializing further, when , that is when is a scalar-valued function of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function . These concepts are named after the
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
Carl Gustav Jacob Jacobi (1804–1851).


Jacobian matrix

The Jacobian of a vector-valued function in several variables generalizes the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a scalar-valued function in several variables, which in turn generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian matrix of a scalar-valued function in several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a single variable is its derivative. At each point where a function is differentiable, its Jacobian matrix can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that the function imposes locally near that point. For example, if is used to smoothly transform an image, the Jacobian matrix , describes how the image in the neighborhood of is transformed. If a function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However a function does not need to be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist. If is differentiable at a point in , then its differential is represented by . In this case, the linear transformation represented by is the best linear approximation of near the point , in the sense that :\mathbf f(\mathbf x) - \mathbf f(\mathbf p) = \mathbf J_(\mathbf p)(\mathbf x - \mathbf p) + o(\, \mathbf x - \mathbf p\, ) \quad (\text \mathbf \to \mathbf), where is a
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
that approaches zero much faster than the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between and does as approaches . This approximation specializes to the approximation of a scalar function of a single variable by its Taylor polynomial of degree one, namely :f(x) - f(p) = f'(p) (x - p) + o(x - p) \quad (\text x \to p). In this sense, the Jacobian may be regarded as a kind of " first-order derivative" of a vector-valued function of several variables. In particular, this means that the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a scalar-valued function of several variables may too be regarded as its "first-order derivative". Composable differentiable functions and satisfy the chain rule, namely \mathbf_(\mathbf) = \mathbf_(\mathbf(\mathbf)) \mathbf_(\mathbf) for in . 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 , then is a function from to itself and the Jacobian matrix is a square matrix. We can then form its
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of near that point. For instance, the continuously differentiable function is invertible near a point if the Jacobian determinant at is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at is positive, then preserves orientation near ; if it is negative, reverses orientation. The absolute value of the Jacobian determinant at gives us the factor by which the function expands or shrinks
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
s near ; 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 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. This is because the -dimensional element is in general a parallelepiped in the new coordinate system, and the -volume of a parallelepiped is the determinant of its edge vectors. The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point. Its applications include determining the stability of the disease-free equilibrium in disease modelling.


Inverse

According to the inverse function theorem, the
matrix inverse In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplic ...
of the Jacobian matrix of an invertible function is the Jacobian matrix of the ''inverse'' function. That is, if the Jacobian of the function is continuous and nonsingular at the point in , then is invertible when restricted to some neighborhood of and :\mathbf J_ = ^ . In other words, if the Jacobian determinant is not zero at a point, then the function is ''locally invertible'' near this point, that is, there is a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of this point in which the function is invertible. The (unproved)
Jacobian conjecture In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero c ...
is related to global invertibility in the case of a polynomial function, that is a function defined by ''n''
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s 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 is a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, a ''critical point'' of 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 be the maximal dimension of the open balls contained in the image of ; then a point is critical if all minors of rank of are zero. In the case where , a point is critical if the Jacobian determinant is zero.


Examples


Example 1

Consider the function with given by : \mathbf f\left(\begin x\\y\end\right) = \begin f_1(x,y)\\f_2(x,y)\end = \begin x^2 y \\5 x + \sin y \end. Then we have :f_1(x, y) = x^2 y and :f_2(x, y) = 5 x + \sin y and the Jacobian matrix of is :\mathbf J_(x, y) = \begin \dfrac & \dfrac\\ em \dfrac & \dfrac \end = \begin 2 x y & x^2 \\ 5 & \cos y \end and the Jacobian determinant is :\det(\mathbf J_(x, y)) = 2 x y \cos y - 5 x^2 .


Example 2: polar-Cartesian transformation

The transformation from
polar coordinates In 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 reference point and an angle from a reference direction. The reference point (analogous to th ...
to
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(''x'', ''y''), is given by the function with components: :\begin x &= r \cos \varphi ; \\ y &= r \sin \varphi . \end :\mathbf J_(r, \varphi) = \begin \dfrac & \dfrac\\ em \dfrac & \dfrac \end = \begin \cos\varphi & - r\sin \varphi \\ \sin\varphi & r\cos \varphi \end The Jacobian determinant is equal to . This can be used to transform integrals between the two coordinate systems: :\iint_ f(x, y) \,dx \,dy = \iint_A f(r \cos \varphi, r \sin \varphi) \, r \, dr \, d\varphi .


Example 3: spherical-Cartesian transformation

The transformation from spherical coordinates to
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(''x'', ''y'', ''z''), is given by the function with components: :\begin x &= \rho \sin \varphi \cos \theta ; \\ y &= \rho \sin \varphi \sin \theta ; \\ z &= \rho \cos \varphi . \end The Jacobian matrix for this coordinate change is :\mathbf J_(\rho, \varphi, \theta) = \begin \dfrac & \dfrac & \dfrac \\ em \dfrac & \dfrac & \dfrac \\ em \dfrac & \dfrac & \dfrac \end = \begin \sin \varphi \cos \theta & \rho \cos \varphi \cos \theta & -\rho \sin \varphi \sin \theta \\ \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & - \rho \sin \varphi & 0 \end. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
is . Since is the volume for a rectangular differential volume element (because the volume of a rectangular prism is the product of its sides), we can interpret as the volume of the spherical differential volume element. Unlike rectangular differential volume element's volume, this differential volume element's volume is not a constant, and varies with coordinates ( and ). It can be used to transform integrals between the two coordinate systems: :\iiint_ f(x, y, z) \,dx \,dy \,dz = \iiint_U f(\rho \sin \varphi \cos \theta, \rho \sin \varphi\sin \theta, \rho \cos \varphi) \, \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta .


Example 4

The Jacobian matrix of the function with components :\begin y_1 &= x_1 \\ y_2 &= 5 x_3 \\ y_3 &= 4 x_2^2 - 2 x_3 \\ y_4 &= x_3 \sin x_1 \end is :\mathbf J_(x_1, x_2, x_3) = \begin \dfrac & \dfrac & \dfrac \\ em \dfrac & \dfrac & \dfrac \\ em \dfrac & \dfrac & \dfrac \\ em \dfrac & \dfrac & \dfrac \end = \begin 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8 x_2 & -2 \\ x_3\cos x_1 & 0 & \sin x_1 \end. This example shows that the Jacobian matrix need not be a square matrix.


Example 5

The Jacobian determinant of the function with components :\begin y_1 &= 5x_2 \\ y_2 &= 4x_1^2 - 2 \sin (x_2x_3) \\ y_3 &= x_2 x_3 \end is :\begin 0 & 5 & 0 \\ 8 x_1 & -2 x_3 \cos(x_2 x_3) & -2 x_2 \cos (x_2 x_3) \\ 0 & x_3 & x_2 \end = -8 x_1 \begin 5 & 0 \\ x_3 & x_2 \end = -40 x_1 x_2. From this we see that reverses orientation near those points where and have the same sign; the function is locally invertible everywhere except near points where or . Intuitively, if one starts with a tiny object around the point and apply to that object, one will get a resulting object with approximately times the volume of the original one, with orientation reversed.


Other uses


Regression and least squares fitting

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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
of the form \dot = F(\mathbf), where \dot is the (component-wise) derivative of \mathbf with respect to the
evolution parameter In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
t (time), and F \colon \mathbb^ \to \mathbb^ is differentiable. If F(\mathbf_) = 0, then \mathbf_ is a stationary point (also called a steady state). By the Hartman–Grobman theorem, the behavior of the system near a stationary point is related to the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of \mathbf_ \left( \mathbf_ \right), the Jacobian of F at the stationary point. 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 square 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

*
Center manifold In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modellin ...
* Hessian matrix * Pushforward (differential)


Notes


References


Further reading

* *


External links

*
Mathworld
A more technical explanation of Jacobians {{Matrix classes Multivariable calculus Differential calculus Generalizations of the derivative Determinants Matrices