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 Jacob Jacobi. The term "Jacobian" is normally pronEng|jəˈkoʊbiən, but sometimes also IPA|/dʒəˈkoʊbiən/.

Jacobian matrix

The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function doesn't need to be differentiable for the Jacobian to be defined, since only the partial derivatives are required to exist.

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 the derivative of a multivariate function. For a function of "n" variables, "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 "J" of "F", as follows:

:

This matrix is also denoted by $J\_F(x\_1,ldots,x\_n)$ and $frac\{partial(y\_1,ldots,y\_m)\}\{partial(x\_1,ldots,x\_n)\}$.

The "i" th row ("i" = 1, ..., "m") of this matrix is the gradient of the "i^th" component function "y"_"i": $left(\; abla\; y\_i\; ight)$.

If p is a point in R^"n" and "F" is differentiable at p, then its derivative is given by "J_F"(p). 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

:$F(mathbf\{x\})\; =\; F(mathbf\{p\})\; +\; J\_F(mathbf\{p\})(mathbf\{x\}-mathbf\{p\})\; +\; o(|mathbf\{x\}-mathbf\{p\}|)$

for x close to p and where "o" is the little o-notation.

The Jacobian of the gradient is the Hessian matrix.

Examples

Example 1. The transformation from spherical coordinates to Cartesian coordinates is given by the function "F" : R⁺ × [0,π) × [0,2π) → R³ 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

:

Example 2. The Jacobian matrix of the function "F" : R³ → R⁴ 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

:

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

Jacobian determinant

If "m" = "n", then "F" is a function from "n"-space to "n"-space and the Jacobian 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 a point p ∈ R^"n" 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:$y\_1\; =\; 5x\_2\; ,$:$y\_2\; =\; 4x\_1^2\; -\; 2\; sin\; (x\_2x\_3)\; ,$:$y\_3\; =\; x\_2\; x\_3\; ,$

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.

ee also

* Pushforward (differential)
* Hessian matrix

References

External links

* [http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node77.html Ian Craw's Undergraduate Teaching Page] An easy to understand explanation of Jacobians
* [http://mathworld.wolfram.com/Jacobian.html Mathworld] A more technical explanation of Jacobians