Jacobian

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This page is about the concept in vector calculus. For the French Revolution faction, see Jacobin. For the followers of James II of England and VII of Scotland, see Jacobitism. For other uses see Jacobean.

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.

In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.

These concepts are all named after the mathematician Carl Gustav Jacob Jacobi. The term "Jacobian" is normally pronounced /jəˈkoʊbiən/, but sometimes also /dʒəˈkoʊbiən/.

Contents

Jacobian matrix

The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. 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.

Its importance 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. For a function of n variables, n > 1, the derivative of a numerical function must be matrix-valued, or a partial derivative.

Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y1(x1,...,xn), ..., ym(x1,...,xn). 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)}.

The i th row (i = 1, ..., m) of this matrix is the gradient of yi transposed: \left(\nabla y_i\right)^T.

If p is a point in Rn and F is differentiable at p, then its derivative is given by JF(p) (and this is the easiest way to compute the derivative[citation needed]). In this case, the linear map described by JF(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.

The Jacobian of the gradient is the Hessian matrix.

If the components of F are arranged into a column vector

\mathbf{y} = (y_1, \dots, y_m)

the Jacobian may be represented as an outer product between the del operator and \mathbf{y}:

\frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)} = \nabla \otimes \mathbf{y}

where the outer product symbol is often left out, it being understood that the gradient of a column vector is a matrix.

Examples

The transformation from spherical coordinates to Cartesian coordinates is given by the function F : R+ × [0,π) × [0,2π) → R3 with components:

x_1 = r \sin\phi \cos\theta \,
x_2 = r \sin\phi \sin\theta \,
x_3 = r \cos\phi \,

The Jacobian matrix for this coordinate change is

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

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.

In dynamical systems

Consider a dynamical system of the form x' = F(x), with F : Rn → Rn. If F(x0) = 0, then x0 is a stationary point. The behavior of the system near a stationary point can often be determined by the eigenvalues of JF(x0), the Jacobian of F at the stationary point.[1]

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 also called the "Jacobian" in some sources.

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 p 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.

Example

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

y_1 = 5x_2 \,
y_2 = 4x_1^2 - 2 \sin (x_2x_3) \,
y_3 = x_2 x_3 \,

is

\begin{vmatrix} 0 & 5 & 0 \\ 8x_1 & -2x_3\cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\ 0 & x_3 & x_2 \end{vmatrix}=-8x_1\cdot\begin{vmatrix} 5 & 0\\ x_3&x_2\end{vmatrix}=-40x_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. 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 about 40 times the volume of the original one.

Uses

The Jacobian determinant is used when making a change of variables when integrating a function over its domain. To accommodate for the change of coordinates the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of coordinates is 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.

See also

References

  1. ^ D.K. Arrowsmith and C.M. Place, Dynamical Systems, Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.

External links