Derive a formula for the volume of a d-dimensional

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

ISBN: 9780201380279 40

Solution for problem 16P Chapter B

An Introduction to Thermal Physics | 1st Edition

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Problem 16P

Derive a formula for the volume of a d-dimensional hypersphere.

Step-by-Step Solution:

Step 1 of 3

Solution 16P

The Gaussian function ${e}^{-{x}^{2}}$ has an antiderivative and if it is integrated from $-\infty{}$ to $+\infty{}$ , its numerical value is found to be $\sqrt{\pi{}}.$

So, $\int\limits_{-\infty{}}^{+\infty{}}{e}^{-{x}^{2}}dx=\sqrt{\pi{}}$

If the function is squared and double integration is done on it. The square is values as ${I}^{2}=\pi{}$ .

The expression for area of a d-dimensional hypersphere is given as,

$A{\ }_{d}(r)=\frac{2\pi{}{\ }^{d/2}r{\ }^{d-1}}{\Gamma{}(d/2)}$

The volume of segment that is $dr$ unit thick is given by,

$dV{\ }_{d}(r)=\frac{2\pi{}{\ }^{d/2}r{\ }^{d-1}}{\Gamma{}(d/2)}\times{}dr$

Integrating this expression,

$\int\limits_{\ }^{\ }dV{\ }_{d}(r)=\int\limits_{\ }^{\ }\frac{2\pi{}{\ }^{d/2}r{\ }^{d-1}}{\Gamma{}(d/2)}\times{}dr$

$V{\ }_{d}(r)=\frac{2\pi{}{\ }^{d/2}}{\Gamma{}(d/2)}\int\limits_{\ }^{\ }r{\ }^{d-1}dr$

$V{\ }_{d}(r)=\frac{2\pi{}{\ }^{d/2}}{\Gamma{}(d/2)}\times{}\frac{{r}^{d}}{d}$

$V{\ }_{d}(r)=\frac{\pi{}{\ }^{d/2}{r}^{d}}{\frac{d}{2}\Gamma{}(\frac{d}{2})}$

$V{\ }_{d}(r)=\frac{\pi{}{\ }^{d/2}{r}^{d}}{\Gamma{}(\frac{d}{2}+1)}$

This is the formula for the volume of a d-dimensional hypersphere.

Step 2 of 3

Chapter B, Problem 16P is Solved

Step 3 of 3

Textbook: An Introduction to Thermal Physics

Edition: 1

Author: Daniel V. Schroeder

ISBN: 9780201380279

This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1. The full step-by-step solution to problem: 16P from chapter: B was answered by , our top Physics solution expert on 07/05/17, 04:29AM. The answer to “Derive a formula for the volume of a d-dimensional hypersphere.” is broken down into a number of easy to follow steps, and 10 words. This full solution covers the following key subjects: derive, Dimensional, formula, hypersphere, Volume. This expansive textbook survival guide covers 10 chapters, and 455 solutions. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. Since the solution to 16P from B chapter was answered, more than 380 students have viewed the full step-by-step answer.