Jacobian

From Free net encyclopedia

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 Template:IPA or Template:IPA.

1 Jacobian matrix
- 1.1 Example
2 Jacobian determinant
- 2.1 Example
- 2.2 Uses
3 See also
4 External links

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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:

<math>\begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} \end{bmatrix} </math>

This matrix is denoted by

<math>J_F(x_1,\ldots,x_n)</math> or by <math>\frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)}</math>

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

<math>F(\mathbf{x}) \approx F(\mathbf{p}) + J_F(\mathbf{p})\cdot (\mathbf{x}-\mathbf{p})</math>

for x close to p.

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Example

The Jacobian matrix of the function F : R³ → R⁴ with components:

is:

<math>J_F(x_1,x_2,x_3) =\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix} </math>

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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.

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Example

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

is:

<math>\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</math>

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.

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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 a injectivity between the coordinates that determine the domain. The Jacobian determinant, as a result, is usually well defined.

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