Jacobian

The word Jacobian is used for both matrix and determinant. Jacobian has a finite number of functions and the same number of variables. The functions undergo partial derivatives concerning the variables and are arranged in the rows accordingly. Just like matrix, Jacobian matrix is of different types such as square matrix having the same number of rows and columns and rectangular matrix having the same number of rows and columns. 

What is the Jacobian Matrix?

Matrices have a unique representation and are found in different sizes and forms. Matrices can be classified based on ranks, order, and the content of the matrix. In high school, we have come across different types of matrices based on different parameters. Matrix helps us to simplify calculations, even the complicated calculations performed by computers are first broken into matrices and then solved. To understand the Jacobian Matrix, we need to understand the concept of vector calculus and some properties of Matrices.

As mentioned above, the Jacobian matrix is a result of partial derivatives of its functions concerning variables. The word Jacobian is also used for the determinant of the Jacobian matrix. This matrix contains all partial derivatives of vector functions. Now, the question arises, what is the use of the Jacobian matrix? Jacobian is used for various purposes like in finding the transformation of coordinates called Jacobian transformation and differentiation with coordinate transformation.

Characteristics of a Jacobian Matrix

A Jacobian matrix is a matrix that can be of any form and contains a first-order partial derivative for a vector function. The different forms of the Jacobian matrix are rectangular matrices having a different number of rows and columns that are not the same, square matrices having the same number of rows and columns. Given below is a representation of a Jacobian matrix in a more rigorous mathematical sense.

f: Rn → Rm is a function that takes as input the vector x ∈ Rn and produces as output the vector f(x) ∈ Rm. Therefore, the Jacobian matrix J of f is an m×n matrix.

Variable x is usually the entry for the matrix. Knowing this is highly imperative, as this indicates that the function is differentiable at the point x. Being differentiable at a point indicates that the matrix can be mapped and given a geometric and visual approach to understanding the equations at hand. Polar-Cartesian and Spherical-Cartesian are the most important kind of Jacobian matrices. These matrices are extremely important, as they help in the conversion of one coordinate system into another, which proves to be useful in many mathematical and scientific endeavours.

The importance of the Jacobian matrix is critical in all fields of mathematics, science, and engineering. One prime example is in the field of control engineering, where the use of Jacobian matrices allows the local (approximate) linearization of non-linear systems around a given equilibrium point, thus allowing the use of linear systems techniques, such as the calculation of eigenvalues (and thus allowing an indication of the type of the equilibrium point). Jacobian matrices are also used in the estimation of the internal states of non-linear systems in the construction of an extended Kalman filter. Basically, we can conclude by saying that Jacobian matrices maintain a truly unique and important place in the world of matrices!

Jacobian Determinant

If the Jacobian matrix is a square matrix, then the number of rows and columns is same, thus it can be written as m = n, then f is a function from ℝn to itself. From the Jacobian matrix, we can form a determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes called "Jacobian".

The Jacobian determinant at a given point gives important information about the behaviour of f near that point. If the Jacobian determinant at p is non-zero, then the continuously differentiable function f is invertible near a point p ∈ ℝn. This is the inverse function theorem. Moreover, f preserves orientation near p, if the Jacobian determinant at p is positive. Similarly, f reverses orientation, if it is negative. The absolute value of the Jacobian determinant at p occurs in the general substitution rule because it gives us the factor by which the function f expands or shrinks volumes near p. The Jacobian determinant is used when making a change of variables when evaluating multiple integrals of a function over a region within its domain. The magnitude of the Jacobian determinant arises as a multiplicative factor within the integral to accommodate for the change of coordinates. This is because the n-dimensional dV element (parallelepiped in the new coordinate system) and the n-volume of a parallelepiped is the determinant of its edge vectors. The applications of Jacobian matrix include determining the stability of the disease-free equilibrium in disease modelling.

Solved Examples

Question 1.  Let x (u, v) = u2 – v2, y (u, v) = 2 uv. Therefore, find the Jacobian J (u, v).

Solution 1: Given that x (u, v) = u2 – v2 and y (u, v) = 2 uv.

We know that,

\[J(u,v)=\begin{bmatrix}xu & xv \\yu & yv \end{bmatrix}\] and \[J(u,v)=\begin{bmatrix}2u & -2v \\2v & 2u \end{bmatrix}\]

Therefore, J (u, v) = 4u2 + 4v2.