# Multi-dimensional Integral by Change of Variables

• I
• junt
In summary, the conversation discusses how to perform a change of variable in a 4-dimensional integral and whether it is necessary to use all four new variables. The suggested method is to expand the integral and write it as a product of one-dimensional integrals using a diagonalization process. However, it is also mentioned that there is a faster method using a general formula for n-dimensional Gaussian integrals.
junt
Hi All,

$$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$
I can perform the integration of the integral above easily by changing the variable
$$u=x_2+x_1\\ v=x_2-x_1$$
Of course first computing the Jacobian, and integrating over ##u## and ##v##

I am wondering how you perform the change of variable for 4-dimensional integral like below:

$$\int{\exp(\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i))dx_1dx_2dx_3dx_4}$$

Is it something like:
$$x_2-x_1=u \\ x_2+x_1=v \\ x_4-x_3=p \\ x_4+x_3=q$$
Should this be enough? I think one only needs 4 new variables right? Because I was thinking the integral would be easier if I could do something like:
$$x_2-x_1=u \\ x_3-x_2=v \\ x_4-x_3=p \\ x_1-x_4=q$$
Or in general, how do you perform change of variable in multi-dimensional case. Should you generally perform it by doing:
$$u=a_1x_1+a_2x_2+a_3x_3+a_4x_4 \\ v=b_1x_1+b_2x_2+b_3x_3+b_4x_4 \\ p=c_1x_1+c_2x_2+c_3x_3+c_4x_4 \\ q=d_1x_1+d_2x_2+d_3x_3+d_4x_4$$

And what kind of change of variable does one need to perform to perform the integration of such integration easily?

Last edited by a moderator:
In $\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i)$ you have $x_0$ term, so I assume that indices are cyclic and $x_0=x_4$. You have to expand $\sum_{i=1}^{4}(x_{i}-x_{i-1})^2$ and then write it as (Einstein convention used) $A_{ij}x_i x_j$. Notice that as $x_i x_j=x_j x_i$, matrix $\mathbb{A}$ is not unique. Defining $\mathbb{x}=[x_1,x_2,x_3,x_4]^T$ you can write $\sum_{i=1}^{4}(x_{i}-x_{i-1})^2=\mathbb{x}^T \mathbb{A}\mathbb{x}$. Now you have to make diagonalization: $\mathbb{A}=\mathbb{M}^T\mathbb{D}\mathbb{M}$. Your new coordinates that will allow you to write your integral as a product of four one-dimensional integrals are $\mathbb{y}=\mathbb{M}\mathbb{x}$. However you can do it faster as there is general formula for $n$-dimensional gaussian integrals for which you only have to know matrix $\mathbb{A}$ and its determinant. You can find derivation of that formula for example here:
http://www.weylmann.com/gaussian.pdf

junt

## What is a multi-dimensional integral by change of variables?

A multi-dimensional integral by change of variables is a method used in mathematics and physics to calculate integrals over multiple variables. It involves transforming the original variables into new variables, which makes the integral easier to solve.

## Why is the change of variables method useful for multi-dimensional integrals?

The change of variables method can simplify complex integrals by transforming them into integrals over new variables that are easier to integrate. It can also help solve integrals that are impossible to solve using traditional methods.

## How do you choose the appropriate new variables for a multi-dimensional integral?

The appropriate new variables for a multi-dimensional integral are chosen based on the shape of the region of integration. The goal is to choose variables that transform the region into a simpler shape, such as a rectangle or sphere, making the integral easier to solve.

## What are some common examples of multi-dimensional integrals by change of variables?

Some common examples of multi-dimensional integrals by change of variables include calculating the volume of a 3D object, finding the center of mass of a 3D object, and calculating the probability of a multivariate distribution.

## What are the limitations of the change of variables method for multi-dimensional integrals?

The change of variables method may not always be applicable for every type of multi-dimensional integral. It also requires a good understanding of the original integral and the appropriate transformation to choose for the new variables. Additionally, it can become quite complex for higher-dimensional integrals.

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