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.

Solving 2D Gaussian Integral w.r.t x and y and also w.r.t polar coordinates

w.r.t x and y

To find the double integral of $e^{-\frac{x^2}{2\sigma^2} - \frac{y^2}{2\sigma^2}}$ with respect to $x$ and $y$, you need to specify the limits of integration. Since the function involves the Gaussian distribution, it's common to integrate over the entire real line. The result is related to the normalization constant of the 2D Gaussian distribution.

The double integral is given by:

$\iint e^{-\frac{x^2}{2\sigma^2} - \frac{y^2}{2\sigma^2}} \,dx \,dy$

Assuming integration over the entire real line, this double integral can be evaluated, and the result is:

$\iint e^{-\frac{x^2}{2\sigma^2} - \frac{y^2}{2\sigma^2}} \,dx \,dy = 2\pi\sigma^2$

This result is obtained by expressing the double integral as the product of two Gaussian integrals (one for $x$ and one for $y$), each contributing a factor of $\sqrt{2\pi\sigma^2}$, and then multiplying those factors together.

Keep in mind that the specific limits of integration and the context in which this double integral is applied may vary, so make sure to adjust accordingly based on the problem at hand.

w.r.t. polar coordinates

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.

The 2D Gaussian integral involves integrating a Gaussian function over two variables, typically $x$ and $y$. The general form of the 2D Gaussian integral is:

$\iint e^{-\frac{x^2 + y^2}{2\sigma^2}} \,dx \,dy$

Here, $\sigma$ is a parameter representing the standard deviation of the Gaussian distribution. The limits of integration would depend on the specific context of the problem, but commonly, the integration is performed over the entire real line for both $x$ and $y$.

The 2D Gaussian integral can be evaluated by expressing it in polar coordinates. The transformation from Cartesian to polar coordinates is given by $x = r\cos(\theta)$ and $y = r\sin(\theta)$, where $r$ is the radius and $\theta$ is the angle.

The integral then becomes:

$\int_0^{2\pi} \int_0^\infty e^{-\frac{r^2}{2\sigma^2}} \,r \,dr \,d\theta$

This integral simplifies to:

$2\pi \sigma^2$

This result is obtained by recognizing that the radial part of the integral is a standard Gaussian integral, and the angular part yields a factor of $2\pi$.

So, the 2D Gaussian integral evaluates to $2\pi\sigma^2$, which is the area under the 2D Gaussian distribution.

Derive generalized Gaussian Integral from the Special Case of Gaussian Integral for Mu = 0 and Sigma = 1-by-sqrt(2)

Pre-read

What is the integral 'I' with respect to Gaussian distribution?

The integral with respect to a Gaussian distribution, often referred to as the Gaussian integral, is a classical mathematical result. The Gaussian integral is defined as follows:

$\int_{-\infty}^{\infty} e^{-x^2} \,dx = \sqrt{\pi}$

This integral is frequently encountered in probability theory and statistics, especially when dealing with normal (Gaussian) distributions. The integral represents the area under the bell-shaped curve of the standard normal distribution.

It's important to note that the Gaussian integral has a well-known result, but proving it involves more advanced mathematical techniques such as completing the square and complex analysis. The result is widely used in various branches of mathematics, physics, and engineering. If you encounter a Gaussian distribution with mean $\mu$ and variance $\sigma^2$, the integral can be generalized as:

$\int_{-\infty}^{\infty} e^{-(x-\mu)^2/(2\sigma^2)} \,dx = \sqrt{2\pi\sigma^2}$

This generalization involves a change of variable and scaling to account for the mean ($\mu$) and variance ($\sigma^2$).

Fig 1:



Fig 2:



Showing how Generalized Gaussian Integral and Special Case of Gaussian Integral are related:
Fig 3:

What is Jacobian of change of variables?

Question: What is Jacobian of change of variables?

Answer:

In multivariable calculus, the Jacobian of a change of variables is a determinant that represents the local scaling factor of a transformation from one coordinate system to another. It's essential for evaluating multiple integrals after a change of variables.

Here's a breakdown of key concepts:

Change of Variables:

    It involves switching from one set of variables (e.g., x and y) to a new set (e.g., u and v) to simplify a problem.
    This often involves functions like x = g(u, v) and y = h(u, v).

Jacobian Matrix:

    A matrix containing the partial derivatives of the new variables with respect to the original ones: 
| ∂x/∂u ∂x/∂v | 
| ∂y/∂u ∂y/∂v |

Jacobian Determinant:

    The determinant of this matrix, denoted as J(u, v) or ∂(x, y)/∂(u, v).

Role in Integrals:

    When changing variables in a double integral, the Jacobian is used to adjust the area element: 
    ∬f(x, y) dx dy = ∬f(g(u, v), h(u, v)) |J(u, v)| du dv
    It ensures the integral's value remains consistent after the transformation.

Geometric Interpretation:

    The Jacobian's absolute value represents the factor by which a small area element in the u-v plane transforms into a corresponding area element in the x-y plane.

Key Points:

    The Jacobian is a 2x2 determinant for 2D transformations, 3x3 for 3D, and so on.
    If the Jacobian is zero at a point, the transformation might distort or fold space, potentially leading to issues in integration.

Example:

    For polar coordinates (x = r cosθ, y = r sinθ), the Jacobian is r. This means a small square in the r-θ plane transforms into a corresponding sector in the x-y plane with area r times larger.

If x and y are two random variables, what is E[xy]?

Question: If x and y are two random variables, what is E[xy]?

If $x$ and $y$ are two random variables, the expected value of their product $xy$ is given by:

$E[xy] = \sum_{i}\sum_{j} x_iy_j P(X=x_i, Y=y_j)$

In continuous cases, where $x$ and $y$ are continuous random variables, this is expressed as an integral:

$E[xy] = \int\int xy f(x, y) \,dx\,dy$

where $f(x, y)$ is the joint probability density function of $x$ and $y$.

In simpler terms, the expected value of the product of two random variables is found by multiplying each pair of possible values of $x$ and $y$ by the probability of that particular combination occurring, and then summing or integrating over all possible pairs.

It's important to note that if $x$ and $y$ are independent, then $E[xy] = E[x] \cdot E[y]$. If they are not independent, the expected value of the product is influenced by the covariance between $x$ and $y$, and it can be expressed as:

$E[xy] = E[x] \cdot E[y] + \text{cov}(x, y)$

where $\text{cov}(x, y)$ is the covariance between $x$ and $y$.