Jacobian matrix and determinant

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.

The Jacobian can be understood by considering a unit area in the new coordinate space; and examining how that unit area transforms when mapped into xy coordinate space in which the integral is visually understood.

[5][6][7] The process involves taking partial derivatives with respect to the new coordinates, then applying the determinant and hence obtaining the Jacobian.

This function takes a point x ∈ Rn as input and produces the vector f(x) ∈ Rm as output.

The Jacobian matrix, whose entries are functions of x, is denoted in various ways; other common notations include Df,

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.

[a] 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.

When m = 1, that is when f : Rn → R is a scalar-valued function, the Jacobian matrix reduces to the row vector

; this row vector of all first-order partial derivatives of f is the transpose of the gradient of f, i.e.

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).

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.

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.

where o(‖x − p‖) is a quantity that approaches zero much faster than the distance between x and p does as x approaches p. This approximation specializes to the approximation of a scalar function of a single variable by its Taylor polynomial of degree one, namely

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 of a scalar-valued function of several variables may too be regarded as its "first-order derivative".

For instance, the continuously differentiable function f is invertible near a point p ∈ Rn if the Jacobian determinant at p is non-zero.

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.

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.

The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point.

In other words, let k be the maximal dimension of the open balls contained in the image of f; then a point is critical if all minors of rank k of f are zero.

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

By the Hartman–Grobman theorem, the behavior of the system near a stationary point is related to the eigenvalues of

[11] Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point.

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

[12] A square system of coupled nonlinear equations can be solved iteratively by Newton's method.

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

The Jacobian is also used in random matrices, moments, local sensitivity and statistical diagnostics.

A nonlinear map sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.