Jacobian (original) (raw)

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 : Rn → Rm is a function from Euclidean _n_-space to Euclidean _m_-space. Such a function is given by m real-valued component functions, _y_1(_x_1,...,x n), ..., y m(_x_1,...,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:

This matrix is denoted by

The _i_-th 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 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

for x close to p.

Example

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

_y_1 = _x_1

_y_2 = 5_x_3

_y_3 = 4(_x_2)2 - 2_x_3

_y_4 = _x_3sin(_x_1)

is:

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 : R3 → R3 with components

_y_1 = 5_x_2

_y_2 = 4(_x_1)2 - 2sin(_x_2_x_3)

_y_3 = _x_2_x_3

is:From this we see that F reverses orientation near those points where _x_1 and _x_2 have the same sign; the function is locally invertible everywhere except near points where _x_1=0 or _x_2=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.