JACOBIAN

In vector calculus, the 'Jacobian' is shorthand for either the 'Jacobian matrix' or its determinant, the 'Jacobian determinant'.
In algebraic geometry the 'Jacobian' of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.
These concepts are all named after the mathematician Carl Gustav Jacobi. The term "Jacobian" is normally pronounced , but can also be pronounced .

Contents

Jacobian matrix

Examples

In dynamical systems

Jacobian determinant

Example

Uses

See also

References

External links

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. For ''n'' > 1, the derivative of a numerical function must be matrix-valued, or a partial derivative.
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:
:
This matrix is denoted by
:$J\_F(x\_1,ldots,x\_n)$ or by
The ''i''th row of this matrix is given by the transpose of 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
:
for 'x' close to 'p'.
Note that the Jacobian of the gradient is the Hessian matrix.

Examples

The transformation from spherical coordinates to Cartesian coordinates is given by the function ''F'' : 'R' × [0,π] × [0,2π] → 'R'³ with components:
:$x\_1\; =\; r\; sinphi\; cos\; heta\; ,$
:$x\_2\; =\; r\; sinphi\; sin\; heta\; ,$
:$x\_3\; =\; r\; cosphi\; ,$
The Jacobian matrix for this coordinate change is
:
rac{partial x_1}{partial r} & rac{partial x_1}{partial phi} & rac{partial x_1}{partial heta} \[3pt]
rac{partial x_2}{partial r} & rac{partial x_2}{partial phi} & rac{partial x_2}{partial heta} \[3pt]
rac{partial x_3}{partial r} & rac{partial x_3}{partial phi} & rac{partial x_3}{partial heta} \
end{bmatrix}=egin{bmatrix}
sinphi cos heta & r cosphi cos heta & -r sinphi sin heta \
sinphi sin heta & r cosphi sin heta & r sinphi cos heta \
cosphi & -r sinphi & 0
end{bmatrix}.
The Jacobian matrix of the function ''F'' : 'R'³ → 'R'⁴ with components
:$y\_1\; =\; x\_1\; ,$
:$y\_2\; =\; 5x\_3\; ,$
:$y\_3\; =\; 4x\_2^2\; -\; 2x\_3\; ,$
:$y\_4\; =\; x\_3\; sin(x\_1)\; ,$
is
:
rac{partial y_1}{partial x_1} & rac{partial y_1}{partial x_2} & rac{partial y_1}{partial x_3} \[3pt]
rac{partial y_2}{partial x_1} & rac{partial y_2}{partial x_2} & rac{partial y_2}{partial x_3} \[3pt]
rac{partial y_3}{partial x_1} & rac{partial y_3}{partial x_2} & rac{partial y_3}{partial x_3} \[3pt]
rac{partial y_4}{partial x_1} & rac{partial y_4}{partial x_2} & rac{partial y_4}{partial x_3} \
end{bmatrix}=egin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 5 \ 0 & 8x_2 & -2 \ x_3cos(x_1) & 0 & sin(x_1) end{bmatrix}.
This example shows that the Jacobian need not be a square matrix.

In dynamical systems

Consider a dynamical system of the form ''x''' = ''F''(''x''), with ''F'' : 'R'^''n'' → 'R'^''n''. If ''F''(''x''₀) = 0, then ''x''₀ is a stationary point. The behavior of the system near a stationary point can often be determined by the eigenvalues of ''J''_''F''(''x''₀), the Jacobian of ''F'' at the stationary point.^[1]

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 is also called the "Jacobian" in some sources.
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
:$y\_1\; =\; 5x\_2\; ,$
:$y\_2\; =\; 4x\_1^2\; -\; 2\; sin\; (x\_2x\_3)\; ,$
:$y\_3\; =\; x\_2\; x\_3\; ,$
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.

Uses

The Jacobian determinant is used when making a change of variables when integrating a function over its domain. To accommodate for the change of coordinates the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of coordinates is done in a manner which maintains an injectivity between the coordinates that determine the domain. The Jacobian determinant, as a result, is usually well defined.

See also

★ Pushforward (differential)

★ Hessian matrix

References

1. D.K. Arrowsmith and C.M. Place, ''Dynamical Systems'', Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.

External links

★ Ian Craw's Undergraduate Teaching Page An easy to understand explanation of Jacobians

★ Mathworld A more technical explanation of Jacobians