HISTROY OF WORD JACOBIAN The Jacobian matrix was developed by Carl Gustav Jacob Jacobi (1804-1851), a German Jewish mathematician. The Jacobian is a matrix whose entries are first-order partial derivatives define as where the function is given by m real-valued component functions, y1(x1, …,xn), …,ym (x1, …,xn), continuous (smooth with no breaks or gaps) and differentiable (the derivative must exist at the point being evaluated). If m = n, then the Jacobian matrix is a square matrix. This matrix is denoted by JF (x 1, … , xn). Assume we have a change of variable x = x(u, v) and y = y(u, v).
J(u, v) = = .
INTRODUCTION:
Jacobian (or functional determinant), a determinant with elements aik = ∂yi/∂xk where yi = fi(x1, . . ., xn), 1 ≤ i ≤ n, are functions that have continuous partial derivatives in some region Δ. It is denoted by
(An analogue of the formula for differentiation of an inverse function) this assertion finds numerous applications in the theory of implicit functions.
In order for the explicit expression, in the neighborhood of the point , of the functions y1, . . . . ym that are implicitly defined by the equations
(2) Fk (x1. . . .,xn, y1. . .,ym) = 0 I ≤ k ≤ m
to be possible, it is sufficient that the coordinates of M satisfy equations (2), that the functions Fk have continuous partial derivatives, and that the Jacobian
be nonzero at M.
Jacobian Matrix
The Jacobian is a matrix whose entries are first-order partial derivatives defined as where the function is given by m real-valued component functions, y1(x1, …,xn), …,ym (x1, …,xn), continuous and differentiable. If m = n, then the Jacobian matrix is a square matrix. This matrix is denoted by JF (x 1, … , xn).
The concept about the Jacobian is its determinant: Jacobian determinant, |J|. The analysis of the |J| permits one to characterize the behavior of the function around a given point, which has uses in the social sciences.
DETERMINANT:
The Jacobian determinant (often simply called the Jacobian) is the determinant of the Jacobian matrix.
First, |J| is used to test functional dependence, linear and nonlinear, of a set of equations. If |J| = 0, the equations are functionally dependent. If |J| > 0, the equations are functionally independent.
|J| does not determine the functional relationship, linear or nonlinear. Second, if |J| at a given point is different from zero, the function is invertible near that point, that is, an inverse function exists. Then the Jacobian determinant in conjunction with the implicit function theorem can be used to identify changes in an endogenous variable, which may be a choice or optimization variable, as an exogenous variable changes.
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Unlike the Hessian matrix (the square matrix of second-order partial derivatives of a function), the Jacobian can be used to analyze constrained-optimization problems. However, like the Hessian, calculating the |J| becomes laborious as the dimensions of the matrix increase. In addition, the Jacobian is difficult to use with a nonlinear optimization problem, which produces a Jacobian matrix with elements that may not be constant.
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 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.
|J| is used to test functional dependence, linear and nonlinear, of a set of equations.
If |J| = 0, the equations are functionally dependent.
If |J| > 0, the equations are functionally independent.
|J| does not determine the functional relationship, linear or nonlinear.
If |J| at a given point is different from zero, the function is invertible near that point, that is, an inverse function exists.
Example of Jacobian matrix and Jacobian determinant:
r(cos2 + sin2) = r is the determinant of this matrix
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rn → Rm 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 J of F, as follows:
This matrix is also denoted by and . The i th row (i = 1, …, m) of this matrix is the gradient of the ith component function yi: .
These concepts are named after the mathematician Carl Gustav Jacob Jacobi. The term “Jacobian” is normally pronounced , but sometimes also .
Jacobian matrix
The Jacobian of a function describes the orientation of a tangent plane to the function at a given point. In this way, the Jacobian generalizes the gradient of a scalar valued function of multiple variables which itself generalizes the derivative of a scalar-valued function of a scalar. Likewise, the Jacobian can also be thought of as describing the amount of “stretching” that a transformation imposes. For example, if (x2,y2) = f(x1,y1) is used to transform an image, the Jacobian of f, J(x1,y1) describes how much the image in the neighborhood of (x1,y1) is stretched in the x, y, and xy directions.
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.
The importance of the Jacobian 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.
If p is a point in Rn and F is differentiable at p, then its derivative is given by JF(p). 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
for x close to p and where o is the little o-notation (for , not ) and is the distance between x and p.
In a sense, both the gradient and Jacobian are “first derivatives” – the former the first derivative of ascalar function of several variables, the latter the first derivative of a vector function of several variables. In general, the gradient can be regarded as a special version of the Jacobian: it is the Jacobian of a scalar function of several variables.
The Jacobian of the gradient has a special name: the Hessian matrix, which in a sense is the “second derivative” of the scalar function of several variables in question.
Inverse
According to the inverse function theorem, the matrix inverse of the Jacobian matrix of a function is the Jacobian matrix of the inverse function. That is, for some function F : Rn → Rn and a point p in Rn,
It follows that the (scalar) inverse of the Jacobian determinant of a transformation is the Jacobian determinant of the inverse transformation.
Examples
Example 1. The transformation from spherical coordinates (r, θ, φ) to Cartesian coordinates (x1, x2, x3) , is given by the function F : R+ Ã- [0,Ï€) Ã- [0,2Ï€) → R3 with components:
The Jacobian matrix for this coordinate change is
The determinant is r2 sin θ. As an example, since dV = dx1 dx2 dx3 this determinant implies that dV = r2 sin θ dr dθ dφ, where dV is the differential volume element.
Example 2. The Jacobian matrix of the function F : R3 → R4 with components
is
This example shows that the Jacobian need not be a square matrix.
Example 3.
It shows how a Cartesian coordinate system is transformed into a polar coordinate system:
In dynamical systems
Consider a dynamical system of the form x’ = F(x), where x’ is the (component-wise) time derivative of x, and F : Rn → Rn is continuous and differentiable. If F(x0) = 0, then x0 is a stationary point (also called a fixed point). The behavior of the system near a stationary point is related to the eigen values of JF(x0), the Jacobian of F at the stationary point. Specifically, if the eigen values all have a negative real part, then the system is stable in the operating point, if any eigen value has a positive real part, then the point is unstable.
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.
Change of Variable — the Jacobian
Another technique that can sometimes be useful when trying to evaluate a double (or triple etc) integral generalize the familiar method of integration by substitution.
Assume we have a change of variable x = x(u, v) and y = y(u, v). Suppose that the region S’ in the uv – plane is transformed to a region S in the xy – plane under this transformation. Define the Jacobian of the transformation as
J(u, v) = = .
It turns out that this correctly describes the relationship between the element of area dx dy and the corresponding area element du dv.
With this definition, the change of variable formula becomes:
f (x, y) dx dy = f (x(u, v), y(u, v))J(u, v) du dv.
Note that the formula involves the modulus of the Jacobian.
Example 9.9 Find the area of a circle of radius R.
Solution. Let A be the disc centred at 0 and radius R. The area of A is thus dx dy. We evaluate the integral by changing to polar coordinates, so consider the usual transformation x = r cos, y = r sin between Cartesian and polar co-ordinates. We first compute the Jacobian;
= cos, = sin, = – r sin, = r cos.
Thus
J(r,) = = = r(cos2 + sin2) = r.
We often write this result as
dA = dx dy = r dr d
Using the change of variable formula, we have
dx dy = | J(r,)| dr d = r dr d = 2.
We thus recover the usual area of a circle.
Note that the Jacobian J(r,) = r > 0, so we did indeed take the modulus of the Jacobian above.
Example 9.10 Find the volume of a ball of radius 1.
Solution. Let V be the required volume. The ball is the set {(x, y, z) | x2 + y2 + z21}. It can be thought of as twice the volume enclosed by a hemisphere of radius 1 in the upper half plane, and so
V = 2 dx dy
where the region of integration D consists of the unit disc {(x, y) | x2 + y21}. Although we can try to do this integration directly, the natural co-ordinates to use are plane polars, and so we instead do a change of variable first. As in 9.9, if we write x = r cos, y = r sin, we have dx dy = r dr d. Thus
V = 2 dx dy
=
2() r dr d
=
2d dr
=
4 –
=
.
Note that after the change of variables, the integrand is a product, so we are able to do the dr and d parts of the integral at the same time.
And finally, we show that the same ideas work in 3 dimensions. There are (at least) two co-ordinate systems in 3 which are useful when cylindrical or spherical symmetry arises. One of these, cylindrical polars is given by the transformation
x = r cos, y = r sin, z = z,
and the Jacobian is easily calculated as
= r so dV = dx dy dz = r dr d dz.
The second useful co-ordinate system is spherical polars with transformation
x = r sincos, y = r sinsin, z = r cos.
The transformation is illustrated in Fig 9.3.
Figure 9.3: The transformation from Cartesian to spherical polar co-ordinates.
It is easy to check that Jacobian of this transformation is given by
dV = r2sin dr d d = dx dy dz.
Example 9.11 The moment of inertia of a solid occupying the region R, when rotated about the z – axis is given by the formula
I = (x2 + y2) dV.
Calculate the moment of inertia about the z-axis of the solid of unit density which lies outside the cylinder of radius a, inside the sphere of radius 2a, and above the x – y plane.
Solution. Let I be the moment of inertia of the given solid about the z-axis. A diagram of a cross section of the solid is shown in Fig 9.4.
Figure 9.4: Cross section of the right hand half of the solid outside a cylinder of radius a and inside the sphere of radius 2a
We use cylindrical polar co-ordinates (r,, z); the Jacobian gives dx dy dz = r dr d dz, so
I
=
dr drr2 dz
=
2r3 dr.
We thus have a single integral. Using the substitution u = 4a2 – r2, you can check that the integral evaluates to 22a5/5.
Exercise 9.12 Show that
z2 dxdydz =
(where the integral is over the unit ball x2 + y2 + z2 1) first by using spherical polars, and then by doing the z integration first and using plane polars.
Use in other subjects:
The analysis of the |J| permits one to characterize the behavior of the function around a given point, which has uses in the social sciences.
In numerical linear algebra, the Jacobi method is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after German mathematician Carl Gustav Jakob Jacobi. Description
http://en.wikipedia.org/wiki/Jacobi_method
Jacobian transformation http://www.vosesoftware.com/ModelRiskHelp/index.htm#Analysing_and_using_data/Bayesian/The_Jacobian_transformation.htm
The Jacobian transformation is an algebraic method for determining the probability distribution of a variable y that is a function of just one other variable x (i.e. y is a transformation of x) when we know the probability distribution for x.
Let x be a variable with probability density function f(x) and cumulative distribution function F(x);
Let y be another variable with probability density function f(y) and cumulative distribution function F(y);
Let y be related to x by some function such that x and y increase monotonically, then we can equate changes dF(y) and dF(x) together, i.e.:
|f(y)dy| = |f(x)dx|
Rearranging a little, we get:
is known as the Jacobian.
Example
If x = Uniform(0,c) and y = 1/x:
so
so the Jacobian is
which gives the distribution for y:
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.
Unlike the Hessian matrix, the Jacobian can be used to analyze constrained-optimization problems.
Jacobian determinant: Jacobian determinant, |J|. The analysis of the |J| permits one to characterize the behavior of the function around a given point, which has uses in the social sciences.
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