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Jacobian

In vector calculus, 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.

The Jacobian matrix is named after the mathematician Carl Gustav Jacobi; the term "Jacobian" is pronounced as "yah-KO-bee-un".

Suppose F : RnRm 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 of F, as follows:

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

This matrix is denoted by

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

The i-th row of this matrix is given by the gradient of the function yi for i=1,...,m.

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 said derivative). 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

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

Example The Jacobian matrix of the function F : R3R4 with components:

y1 = x1
y2 = 5x3
y3 = 4(x2)2 - 2x3
y4 = x3sin(x1)

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>

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 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 and only 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 : R3R3 with components

y1 = 5x2
y2 = 4(x1)2 - 2sin(x2x3)
y3 = x2x3
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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