Jacobian matrix and determinant

Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential calculus Definitions Derivative Differential (infinitesimal) Differential of a function Generalizations of the derivative Concepts Notation for differentiation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem Rules and identities Sums Products Chain rule Powers Quotients Faà di Bruno's formula
Integral calculus Lists of integrals Definitions Antiderivative Integral Improper integral Riemann integral Lebesgue integration Contour integral Integration by parts discs cylindrical shells substitution trigonometric substitution partial fractions changing order reduction formulae
Series Geometric Arithmetico-geometric Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (Term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector calculus Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem Directional derivative
Multivariable calculus Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian
Specialized calculi Calculus of variations Fractional Malliavin Stochastic

In vector calculus, the Jacobian matrix (/dʒɨˈkoʊbiən/, /jɨˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector-valued function. Specifically, suppose $F : \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a function (which takes as input real n-tuples and produces as output real m-tuples). Such a function is given by m real-valued component functions, $F_1(x_1,\ldots,x_n),\ldots,F_m(x_1,\ldots,x_n)$. The partial derivatives of all these functions with respect to the variables $x_1,\ldots,x_n$ (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix $J$ of $F$, as follows:

$J=\begin{bmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{bmatrix}.$

This matrix, whose entries are functions of $x_1,\ldots,x_n$, is also denoted by $J_F(x_1,\ldots,x_n)$ and $\frac{\partial(F_1,\ldots,F_m)}{\partial(x_1,\ldots,x_n)}$. (Note that some books define 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 $p=(x_1,\ldots,x_n)$, which is a slightly stronger condition than merely requiring that all partial derivatives exist there, then the derivative of F at p is the linear transformation $\mathbb{R}^n \rightarrow \mathbb{R}^m$ represented by the matrix $J_F(x_1,\ldots,x_n)$. This linear transformation is the best linear approximation of the function F near the point p.

In the case $m=n$ the Jacobian matrix will be a square matrix, and its determinant, a function of $x_1,\ldots,x_n$, is the Jacobian determinant of F. It carries important information about the local behavior of F and can be thought of as a local expansion factor for volumes; it is used when performing variable substitutions in multi-variable integrals, since it occurs prominently in the substitution rule for multiple variables.

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

A simple example

Consider the function $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by

$F(x,y)=\begin{bmatrix} x^2 y \\ 5x + \sin(y) \end{bmatrix}.$

Then we have

$F_1(x,y)=x^2 y$

and

$F_2(x,y)=5x + \sin(y)$

and the Jacobian matrix of F is

$J_F(x,y)=\begin{bmatrix} \dfrac{\partial F_1}{\partial x} & \dfrac{\partial F_1}{\partial y}\\ \dfrac{\partial F_2}{\partial x} & \dfrac{\partial F_2}{\partial y} \end{bmatrix}=\begin{bmatrix} 2xy & x^2\\ 5 & \cos(y) \end{bmatrix}$

and the Jacobian determinant is

$\det(J_F(x,y))=2xy \cos(y) - 5x^2.$

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 a scalar-valued multivariable function is the gradient and that of a scalar-valued function of single variable is 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 $(x_2,y_2)=f(x_1,y_1)$ is used to transform an image, the Jacobian of $f$, $J(x_1,y_1)$ describes how the image in the neighborhood of $(x_1,y_1)$ is transformed.

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 in Rⁿ and F is differentiable 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 approximation of F near the point p, in the sense that

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

for x close to p and where o is the little o-notation (for $x\to p$) and $\|\mathbf{x}-\mathbf{p}\|$ 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:

$f(x) = f(p) + f'(p) ( x - p ) + o(x-p).$

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

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 in Rⁿ, then F is invertible when restricted to some neighborhood of p and

$(J_F^{-1})(F(p)) = [ (J_F)(p) ]^{-1}.\$

Uses

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 continuous and differentiable. If F(x₀) = 0, then x₀ is a stationary point (also called a critical point, not to be confused with a fixed point). The behavior of the system near a stationary point is related to the eigenvalues of J_F(x₀), the Jacobian of F at the stationary point. Specifically, if the eigenvalues all have real parts that are less than 0, then the system is stable near the stationary point, if any eigenvalue has a real part that is greater than 0, then the point is unstable. If the largest real part of the eigenvalues is equal to 0, 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.

The following is the detail code in MATLAB (although there is a built in 'jacobian' command)

   function s = newton(f, x, tol)
   % f is a multivariable function handle, x is a starting point, both given as row vectors
   % s is solution of f(s)=0 found by Newton's method
   if nargin == 2
       tol = 10^(-5);
   end
   while 1
       % if x and f(x) are row vectors, we need transpose operations here
       y = x' - jacob(f, x)\f(x)';             % get the next point
       if norm(f(y))<tol                       % check error tolerate
           s = y';
           return;
       end
       x = y';
   end

   function j = jacob(f, x)            % approximately calculate Jacobian matrix
   k = length(x);
   j = zeros(k, k);
   x2 = x;
   dx = 0.001;
   for m = 1: k
       x2(m) = x(m)+dx;
       j(m, :) = (f(x2)-f(x))/dx;      % partial derivatives in m-th row
       x2(m) = x(m);
   end

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 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 ∈ Rⁿ 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.

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 solve systems of differential equations at an equilibrium point or approximate solutions near an equilibrium point.

Further examples

Example 1. The transformation from spherical coordinates (r, θ, φ) to Cartesian coordinates (x₁, x₂, x₃), is given by the function F : R⁺ × [0,π] × [0,2π) → R³ with components:

$x_1 = r\, \sin\theta\, \cos\phi \,$

$x_2 = r\, \sin\theta\, \sin\phi \,$

$x_3 = r\, \cos\theta. \,$

The Jacobian matrix for this coordinate change is

$J_F(r,\theta,\phi) =\begin{bmatrix} \dfrac{\partial x_1}{\partial r} & \dfrac{\partial x_1}{\partial \theta} & \dfrac{\partial x_1}{\partial \phi} \\[3pt] \dfrac{\partial x_2}{\partial r} & \dfrac{\partial x_2}{\partial \theta} & \dfrac{\partial x_2}{\partial \phi} \\[3pt] \dfrac{\partial x_3}{\partial r} & \dfrac{\partial x_3}{\partial \theta} & \dfrac{\partial x_3}{\partial \phi} \\ \end{bmatrix}=\begin{bmatrix} \sin\theta\, \cos\phi & r\, \cos\theta\, \cos\phi & -r\, \sin\theta\, \sin\phi \\ \sin\theta\, \sin\phi & r\, \cos\theta\, \sin\phi & r\, \sin\theta\, \cos\phi \\ \cos\theta & -r\, \sin\theta & 0 \end{bmatrix}.$

The determinant is r² sin θ. As an example, since dV = dx₁ dx₂ dx₃ this determinant implies that the differential volume element dV = r² sin θ dr dθ dϕ. Nevertheless this determinant varies with coordinates.

Example 2. The Jacobian matrix of the function F : R³ → R⁴ 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

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

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

Example 3.

$x\,=r\,\cos\,\phi;$

$y\,=r\,\sin\,\phi.$

$J(r,\phi)=\begin{bmatrix} {\partial x\over\partial r} & {\partial x\over \partial\phi} \\ {\partial y\over \partial r} & {\partial y\over \partial\phi} \end{bmatrix}=\begin{bmatrix} {\partial (r\cos\phi)\over \partial r} & {\partial (r\cos\phi)\over \partial \phi} \\ {\partial(r\sin\phi)\over \partial r} & {\partial (r\sin\phi)\over \partial\phi} \end{bmatrix}=\begin{bmatrix} \cos\phi & -r\sin\phi \\ \sin\phi & r\cos\phi \end{bmatrix}$

The Jacobian determinant is equal to $r$. This shows how an integral in the Cartesian coordinate system is transformed into an integral in the polar coordinate system:

$\iint_A dx\, dy= \iint_B r \,dr\, d\phi.$

Example 4. The Jacobian determinant of the function F : R³ → R³ with components

$\begin{align} y_1 &= 5x_2 \\ y_2 &= 4x_1^2 - 2 \sin (x_2x_3) \\ y_3 &= x_2 x_3 \end{align}$

is

$\begin{vmatrix} 0 & 5 & 0 \\ 8 x_1 & -2 x_3 \cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\ 0 & x_3 & x_2 \end{vmatrix} = -8 x_1 \cdot \begin{vmatrix} 5 & 0 \\ x_3 & x_2 \end{vmatrix} = -40 x_1 x_2.$

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. Intuitively, if you start with a tiny object around the point (1,2,3) and apply F to that object, you will get a resulting object with approximately 40×1×2=80 times the volume of the original one.

See also

• Hessian matrix
• Pushforward (differential)

External links

• Hazewinkel, Michiel, ed. (2001), "Jacobi matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Mathworld A more technical explanation of Jacobians