Jacobian

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

Also, 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.

They are all named after the mathematician Carl Gustav Jacobi; the term "Jacobian" may be pronounced as [ja ˈko bi ən] or [ʤə ˈko bi ən].

Contents

1 Jacobian matrix

1.1 Examples
1.2 The Jacobian matrix in dynamical systems

2 Jacobian determinant

2.1 Example
2.2 Uses

3 See also
4 References
5 External links

Jacobian matrix

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

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 of F, as follows:

[\begin \frac & \cdots & \frac \\ \vdots & \ddots & \vdots \\ \frac & \cdots & \frac \end ]

This matrix is denoted by

[J_F(x_1,\ldots,x_n)] or by [\frac]

The ith row of this matrix is given by the gradient of the function y_i for i=1,...,m.

If p is a point in Rⁿ and F is differentiable at p, then its derivative is given by J_F(p) (and this is the easiest way to compute the derivative ). 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) \approx F(\mathbf) + J_F(\mathbf)\cdot (\mathbf-\mathbf)]

for x close to p.

Examples

Consider spherical polar coordinates. The following function constitutes a change of variables back to cartesian coordinates.

F : R x [0,2π] x [0,π] → R³ with components:

[ x_1 = r \cos\phi \sin\theta \,]

[ x_2 = r \sin\phi \sin\theta \,]

[ x_3 = r \cos\theta \,]

The Jacobian Matrix for this change is

[J_F(r,\phi,\theta) =\begin \cos\phi \sin\theta & -r \sin\phi \sin\theta & r \cos\phi \cos\theta \\ \sin\phi \sin\theta & r \cos\phi \sin\theta & r \sin\phi \cos\theta \\ \cos\theta & 0 & -r \sin\theta \end ]

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 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end ]

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

The Jacobian 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 behaviour of the system near a stationary point is often determined by J_F(x₀), the Jacobian of F at the stationary point.D.K. Arrowsmith and C.M. Place, Dynamical Systems, Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.

Jacobian determinant

If m = n, then F is a function from n-space to n-space and the Jacobi 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 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=-8x_1\cdot\begin 5 & 0\\ x_3&x_2\end=-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 basis the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of basis 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.

References

External links

[Ian Craw's Undergraduate Teaching Page] An easy to understand explanation of Jacobians
[Mathworld] A more technical explanation of Jacobians

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