The Jacobian matrix is named after the mathematician Carl Gustav Jacobi; the term "Jacobian" is pronounced as "yahKObeeun".
Suppose F : R^{n} → R^{m} is a function from Euclidean nspace to Euclidean mspace. Such a function is given by m realvalued 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 mbyn matrix, the Jacobian matrix of F, as follows:
This matrix is denoted by
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 R^{n} and F is differentiable at p, then its derivative is given by J_{F}(p) (and this is the easiest way to compute said 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
Example The Jacobian matrix of the function F : R^{3} → R^{4} with components:
is:
Jacobian determinant If m = n, then F is a function from nspace to nspace 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 nonzero. 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 : R^{3} → R^{3} with components
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.
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