# Integration Over Spheres in R^d

• Mathmajor2010
In summary, The conversation discusses a proof involving a function \phi and a set S, where \phi is constant on spheres and S is defined as x \in \mathbb{R}^d : ||x|| = 1. The conversation also mentions a coordinate transformation and the use of a Jacobian determinant to explain the presence of t^{d-1} in the second integral.
Mathmajor2010

## Homework Statement

I'm a bit confused at a single step in a proof. Let $\phi \in L^1(\mathbb{R}) \cap C(\mathbb{R}^d)$ be a function such that for any $\omega \in \mathbb{R}^d$, $\phi(\omega) = \psi(||\omega)||$. That is, the function depends solely on the norm of the vector input, so it is constant on spheres I suppose.

Let $S = \{ x \in \mathbb{R}^d : ||x|| = 1\}$. Then, we have
$$\int_{\mathbb{R}^d} \phi(\omega) e^{-ix^T \omega} d\omega = \int_0^{\infty} t^{d-1} dt \int_{S} \phi(t ||\omega||) e^{-ix^t \omega} dS(\omega)$$

I'm not sure exactly what they did to jump from the first integral to the second. I understand the "idea" is that since the function is constant on spheres, simply integrate on the sphere of radius t and then integrate as t goes over all positive numbers, but I'm not sure how they got that $t^{d-1}$ . It's been a while since I've had a multivariable calculus class, so I'm not sure what I'm missing. I assume this is some coordinate transformation and the function of t comes out of a Jacobian of some sort, but I'm not exactly sure. Any suggestions in the right direction would be great. Thanks!

I don't quite understand as the sphere S you have defined is d-1 dimensional.

As an aside the $t^{d-1}$ comes from the Jacobian

## 1. What is integration over spheres in R^d?

Integration over spheres in R^d refers to the process of calculating the volume or surface area of a sphere in a d-dimensional space. It involves using mathematical techniques to find the integral of a function over a spherical region.

## 2. Why is integration over spheres in R^d important?

Integration over spheres in R^d is important in many areas of science and mathematics, such as physics, engineering, and statistics. It allows us to calculate important quantities like the volume of a sphere, the surface area of a sphere, and the average value of a function over a spherical region.

## 3. What are some techniques for integrating over spheres in R^d?

Some techniques for integrating over spheres in R^d include using spherical coordinates, which involve converting the integral to a triple integral and using special formulas for the Jacobian; using the divergence theorem, which relates the integral over a sphere to the integral of the divergence of the function over the entire region; and using symmetry arguments to simplify the integral.

## 4. How is integration over spheres in R^d different from integration over other shapes?

The main difference between integration over spheres in R^d and integration over other shapes is the use of spherical coordinates. Spherical coordinates are unique to spheres and involve using different variables (radius, inclination, and azimuth) to represent the spherical region. This can make the integration more complicated compared to other shapes, but also allows for more efficient and accurate calculations.

## 5. Can integration over spheres in R^d be applied to real-world problems?

Yes, integration over spheres in R^d can be applied to real-world problems in various fields like physics, engineering, and statistics. For example, it can be used to calculate the volume and surface area of a spherical object, such as a planet or a cell, or to find the average value of a function over a spherical region, which can be useful in analyzing data. It is a fundamental concept in mathematics and has many practical applications.

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