Understanding Jacobian: Jacobian Matrix, Determinant and Examples

Understanding the Jacobian

The term 'Jacobian' can refer to both the Jacobian matrix and the Jacobian determinant. The matrix is defined for a finite number of functions with an equal number of variables. Each row of the matrix consists of the first partial derivative of a function with respect to its variables. The Jacobian matrix can be square (equal number of rows and columns) or rectangular (unequal number of rows and columns).

Defining the Jacobian Matrix

Let's consider a function f: ℝ³ → ℝ. The derivative at p for a row vector can be defined as:

The Jacobian matrix for this function is:

The determinant of this Jacobian matrix is referred to as the Jacobian determinant.

Jacobian Determinant

Let's consider a function f: ℝ³ → ℝ defined as f (x, y) = (u (x, y), v (x, y)). The Jacobian matrix for this function is:

The determinant of this Jacobian matrix is:

Transformation from Cartesian to Polar Coordinates

In a typical transformation from cartesian to polar coordinates, the equations are:

x = r cos θ

y = r sin θ

The Jacobian determinant for this transformation is:

Applying these partial differentiations to the polar equations, we get:

J (r, θ ) = r (sin²θ + cos²θ) = r (1)

J (r, θ) = 1

Example of Jacobian

Problem: Suppose x (u, v) = u² - v² and y (u, v) = 2 uv. Can you determine the Jacobian J (u, v)?

Solution:

Given: x (u, v) = u² - v²

y (u, v) = 2 uv

The Jacobian is:

J (u, v) = 4u² + 4v²

Therefore, J (u, v) equals 4u² + 4v².