How to calculate the volume of this space

  • Thread starter wdlang
  • Start date
  • #1
307
0

Main Question or Discussion Point

in n-dimensional space, there is an area defined by

\sum_i x_i^4 <1

what is its volume?
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
250
Hi wdlang! :smile:

(try using the X2 and X2 tags just above the Reply box :wink:)

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 
  • #3
193
0
Hi all!

I would suggest you give it a try using the transformation formula!

The following maps could be helpful:

[tex]R^n\rightarrow R^n_{\geq 0}\rightarrow B^n[/tex], where B^n is the unit ball.

[tex](x_1,...,x_n)\rightarrow (x_1^2,...,x_n^2):=(y_1,...,y_n)\rightarrow ||y||_2^2=\displaystyle\sum_{k=1}^n y_k^2=\displaystyle\sum_{k=1}^n x_k^4[/tex]


Good luck!
 
  • #4
307
0
Hi all!

I would suggest you give it a try using the transformation formula!

The following maps could be helpful:

[tex]R^n\rightarrow R^n_{\geq 0}\rightarrow B^n[/tex], where B^n is the unit ball.

[tex](x_1,...,x_n)\rightarrow (x_1^2,...,x_n^2):=(y_1,...,y_n)\rightarrow ||y||_2^2=\displaystyle\sum_{k=1}^n y_k^2=\displaystyle\sum_{k=1}^n x_k^4[/tex]


Good luck!
I will try that

i am a physics student, so i am satisfied with a numerical value

so i will also try it with Monte Carlo simulation
 

Related Threads on How to calculate the volume of this space

Replies
5
Views
4K
  • Last Post
Replies
6
Views
806
  • Last Post
Replies
8
Views
11K
Replies
8
Views
12K
  • Last Post
Replies
2
Views
4K
Replies
1
Views
758
Replies
9
Views
22K
Replies
6
Views
2K
  • Last Post
Replies
3
Views
6K
Replies
6
Views
1K
Top