Derive Jacobian when we are changing from 2D cartesian coordinates to polar coordinates

Here's the derivation of the Jacobian when changing from 2D Cartesian coordinates to polar coordinates:

1. Transformation Equations:

• Cartesian coordinates (x, y) are related to polar coordinates (r, θ) by:
  • x = r cos(θ)
  • y = r sin(θ)

2. Jacobian Matrix:

• The Jacobian matrix represents the partial derivatives of the new coordinates with respect to the old coordinates:

  J = | ∂x/∂r ∂x/∂θ | | ∂y/∂r ∂y/∂θ |

3. Calculating Partial Derivatives:

• ∂x/∂r = cos(θ)
• ∂x/∂θ = -r sin(θ)
• ∂y/∂r = sin(θ)
• ∂y/∂θ = r cos(θ)

4. Filling the Jacobian Matrix:

J = | cos(θ) -r sin(θ) | | sin(θ) r cos(θ) |

5. Evaluating the Determinant:

• The Jacobian, denoted as J, is the determinant of this matrix:

  J = det(J) = cos(θ) * r cos(θ) - (-r sin(θ)) * sin(θ) = r cos^2(θ) + r sin^2(θ) = r (cos^2(θ) + sin^2(θ))

6. Simplifying Using Trigonometric Identity:

• Since cos^2(θ) + sin^2(θ) = 1, we get:

  J = r

Therefore, the Jacobian for the change of variables from Cartesian coordinates to polar coordinates is r.

Significance in Integration:

• When changing variables in a double integral from Cartesian to polar coordinates, the area element dx dy transforms to r dr dθ. This is a crucial step in evaluating integrals over regions that are more naturally expressed in polar coordinates, such as circles and sectors.

Using ChatGPT

Let's go through the steps to calculate the Jacobian for the transformation from Cartesian to polar coordinates.

The transformation from Cartesian coordinates $x$ and $y$ to polar coordinates $r$ and $\theta$ is given by:

$x = r \cos(\theta)$ $y = r \sin(\theta)$

The Jacobian of this transformation is the determinant of the matrix of partial derivatives:

$J = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix}$

1. Partial Derivatives: Calculate the partial derivatives:

   $\frac{\partial x}{\partial r} = \cos(\theta)$ $\frac{\partial x}{\partial \theta} = -r \sin(\theta)$ $\frac{\partial y}{\partial r} = \sin(\theta)$ $\frac{\partial y}{\partial \theta} = r \cos(\theta)$

2. Jacobian Determinant: Compute the determinant:

   $J = \cos(\theta) \cdot r \cos(\theta) - (-r \sin(\theta)) \cdot \sin(\theta)$ $J = r \cos^2(\theta) + r \sin^2(\theta)$ $J = r$

So, the Jacobian determinant is $r$ for the given transformation. When changing variables in an integral, you need to multiply by the absolute value of the Jacobian determinant. In polar coordinates, this means you multiply by $r$. The $dx\,dy$ becomes $r\,dr\,d\theta$ in polar coordinates.