Jacobian matrix and determinant

In vector calculus, the Jacobian matrix ( rarely /jɨˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rⁿ → R^m is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y₁(x₁,...,x_n), ..., y_m(x₁,...,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::

\( J=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix}. \)

This matrix is also denoted by \( J_F(x_1,\ldots,x_n) \) and \( \frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)} \) .If (x₁,...,x_n) are the usual orthogonal Cartesian coordinates, the i th row (i = 1, ..., m) of this matrix corresponds to the gradient of the i^th component function y_i: \( \left(\nabla y_i\right) \). 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 determinant of the Jacobian matrix (if m=n).

These concepts are named after the mathematician Carl Gustav Jacob Jacobi.

Jacobian matrix

The Jacobian of a function describes the orientation of a tangent plane to the function at a given point. In this way, 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 scalar. Likewise, the Jacobian can also be thought of as describing the amount of "stretching" that a transformation imposes. 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 much the image in the neighborhood of \( (x_1,y_1) \) 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 derivatives 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 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.

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. In general, the gradient 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 gradient has a special name: the Hessian matrix, which in a sense is the "second derivative" of the scalar function of several variables 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, for some function F : Rⁿ → Rⁿ and a point p in Rⁿ,,

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

It follows that the (scalar) inverse of the Jacobian determinant of a transformation is the Jacobian determinant of the inverse transformation.
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.^[1] 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.

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

function s = jacobian(f, x, tol)
% f is a multivariable function handle, x is a starting point
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);
for m = 1: k
x2 = x;
x2(m) =x(m)+0.001;
j(m, :) = 1000*(f(x2)-f(x)); % partial derivatives in m-th row
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.
Uses

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. 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. The Jacobian can also be used to solve systems of differential equations at an equilibrium point or approximate solutions near an equilibrium point.
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. To avoid any variation the new coordinates can be defined as \( w_{1}=\frac{r^{3}}{3},\ w_{2}=-\cos\theta,\ w_{3}=\phi.\, [2] \) Now the determinant equals to 1 and volume element becomes \( r^{2}dr\ \sin\theta\ d\theta\ d\phi=dw_{1}dw_{2}dw_{3}\,. \)

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 (1,1,1) and apply F to that object, you will get an object set with approximately 40 times the volume of the original one.
See also

Pushforward (differential)
Hessian matrix

Notes

^ D.K. Arrowsmith and C.M. Place, Dynamical Systems, Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.
^ Taken from http://www.sjcrothers.plasmaresources.com/schwarzschild.pdf - On the Gravitational Field of a Mass Point according to Einstein’s Theory by K. Schwarzschild - arXiv:physics/9905030 v1 (text of the original paper, in Wikisource).

External links

Mathworld A more technical explanation of Jacobians

Mathematics Encyclopedia