# Jacobian

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/.

## Contents

## 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* : **R**^{n} → **R**^{m} 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:

**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 *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 **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 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

**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* : **R**^{3} → **R**^{4} with components:

*y*_{1}=*x*_{1}*y*_{2}= 5*x*_{3}*y*_{3}= 4(*x*_{2})^{2}− 2*x*_{3}*y*_{4}=*x*_{3}sin(*x*_{1})

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* : **R**^{3} → **R**^{3} 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:

**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 *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.

## See also

## External links

- Ian Craw's Undergraduate Teaching Page An easy to understand explanation of Jacobians
- Mathworld A more technical explanation of Jacobians

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