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 structure, which can be imposed on the curve.

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

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 : 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

This matrix is denoted by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)}}

The ith 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 the 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\mathbf{x}) \approx F(\mathbf{p}) + J_F(\mathbf{p})\cdot (\mathbf{x}-\mathbf{p})}

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }

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 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}

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.

See also

External links

ar:مصفوفة جاكوبي de:Jacobi-Matrix fr:Matrice jacobienne it:Matrice jacobiana he:יעקוביאן ja:関数行列 pl:Jakobian